TODAY Show: Tech companies changing face of education

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Language: en

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DAVID GREGORY: Back now at 8:36
with Education Nation Today.

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This morning, how
technology is leading

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to a new kind of learning.

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NBC's Craig Melvin
is here with a look

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at the classroom of the future.

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Hey, Craig, good morning.

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CRAIG MELVIN: David,
good morning to you.

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Savannah, good morning
to you, as well.

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From text books to
tablets, have you

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ever wondered what
our classrooms

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will look like 10, 20,
even 50 years from now?

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Well, some innovative
thinkers are

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unlocking new and powerful
ways to teach our kids,

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changing the way we look
at our education system

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in the process.

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The numbers alone
are staggering.

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JOSE FERREIRA: The thing
about the education system

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is, it's a very
impersonal system.

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It's a factory model.

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CRAIG MELVIN: One student
drops out of high school

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every 26 seconds.

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Roughly 30% of kids
each year will not

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graduate from high school.

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That's 1.3 million
in all, falling

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through the cracks of
our education system.

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Worldwide, the numbers
grow exponentially.

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A whopping 80% of kids will
not get a high school diploma.

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Experts say each of those
kids is a lost opportunity.

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How many DaVincis and Einsteins
and Marie Curies and Michael

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Jordans are we losing
every generation

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because we're just
not giving them

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the opportunities
that some of us have.

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CRAIG MELVIN: But what
if that could be changed?

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What if an internet
connection was the difference

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in getting a world
class education.

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Today, education is having
that internet moment.

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And a chosen few are taking
some bold steps forward.

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JOSE FERREIRA: I'm Jose
Ferreira, the founder

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and CEO of Newton.

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We're trying to revolutionize
education across the globe.

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CRAIG MELVIN: Newton is an
adaptive learning platform

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that personalizes education
based on each kid's strengths

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and weaknesses.

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In essence, students
go at their own pace

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and the software continuously
adapts to their learning style.

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JOSE FERREIRA: If you learn
math best with medium difficulty

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practice questions,
we know that.

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If you learn science best
in 24-minute bite sizes,

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we know that.

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Everybody has a
learning experience

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that is unique to them.

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CRAIG MELVIN: Ferreira
started the company in 2008,

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using his credit card.

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Since then, Newton has
grown by leaps and bounds.

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They currently reach almost
five million students,

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and that number is growing.

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JOSE FERREIRA: We
want to end the access

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problem for education.

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We want every kid, whether
you're in the inner city

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or in the developing
world, we want

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you to have the best possible
education you can get.

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SAL KHAN: My name
is Sal Khan and I'm

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the founder of the Khan Academy.

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CRAIG MELVIN: In 2004, Sal
Kahn was tutoring his cousins

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remotely from his
then home in Boston.

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The lessons started
over the phone,

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until a friend
suggested he upload

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educational videos to YouTube.

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SAL KHAN: And I
immediately said, no.

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Videos and YouTube, that's
for cats playing piano,

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not serious mathematics.

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But I gave it a shot.

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And those videos took
on a life of their own.

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CRAIG MELVIN: Those videos
became the cornerstone

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of Khan Academy,
a virtual library

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of now more than 4,000
tutorials covering everything

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from basic math to
college level chemistry.

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SAL KHAN: And our goal over
the next 5, 10, 50, 500 years

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is to go from the six
million students that we're

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serving every month now to 60
million, 600 million students.

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CRAIG MELVIN: One of those
students recently sent Khan

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a video message from the
unlikeliest of places,

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an orphanage in Mongolia.

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STUDENT: Hey, Sal.

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Your lessons are so
interesting and funny.

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Make more lessons.

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SAL KHAN: That by
itself is pretty mind

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blowing for us, this idea that
the work that we get to do

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can reach people as far as
an orphanage in Mongolia.

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CRAIG MELVIN: Two companies
led by two visionaries, both

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using the reach of
technology to reimagine

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the future of education.

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And now Newton
recently partnered

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with education publisher
Houghton Mifflin Harcourt.

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As a result, more
kids will have access

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to their adaptive
learning platform.

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And we should note, Khan
Academy's mission statement,

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changing education
for the better

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by providing a free, world
class education for anyone,

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anywhere, that means all
of their tutorial videos

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are free of charge.

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SAVANNAH GUTHRIE:
That is pretty cool.

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DAVID GREGORY: Great stuff.

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Great model.

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CRAIG MELVIN: Thank you.

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Good to see you.

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I'm looking forward to seeing
you dance in a few minutes,

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too.

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DAVID GREGORY: Well,
you never know.


Recognizing rational and irrational numbers (examples)

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Which of the following real
numbers are irrational?

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Well, irrational just
means it's not rational.

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It means that you cannot
express it as the ratio of two

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integers.

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So let's see what we have here.

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So we have the square
root of 8 over 2.

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If you take the square root
of a number that is not

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a perfect square, it is
going to be irrational.

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And then if you just take
that irrational number

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and you multiply it, and you
divide it by any other numbers,

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you're still going to
get an irrational number.

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So square root of
8 is irrational.

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You divide that by 2,
it is still irrational.

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So this is not rational.

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Or in other words, I'm
saying it is irrational.

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Now, you have pi,
3.14159-- it just

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keeps going on and on and on
forever without ever repeating.

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So this is irrational,
probably the most famous

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of all of the
irrational numbers.

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5.0-- well, I can
represent 5.0 as 5/1.

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So 5.0 is rational.

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It is not irrational.

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0.325-- well, this is the
same thing as 325/1000.

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So I can clearly represent
it as a ratio of integers.

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So this is rational.

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Just as I could represent
5.0 as 5/1, both of these

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are rational.

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They are not irrational.

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Here I have
7.777777, and it just

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keeps going on and
on and on forever.

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And the way we denote
that, you could just

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say these dots that say
that the 7's keep going.

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Or you could say 7.7.

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And this line shows that
the 7 part, the second 7,

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just keeps repeating on forever.

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Now, if you have a repeating
decimal-- in other videos,

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we'll actually convert
them into fractions--

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but a repeating decimal
can be represented

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as a ratio of two integers.

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Just as 1/3 is equal to
0.333 on and on and on.

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Or I could say it like this.

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I could say 3 repeating.

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We can also do the
same thing for that.

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I won't do it here,
but this is rational.

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So it's not irrational.

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8 and 1/2?

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Well, that's the same thing.

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8 and 1/2 is the
same thing as 17/2.

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So it's clearly rational.

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So the only two
irrational numbers

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are the first two
right over here.


Percent word problem example 3

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A zoo has 15
emperor penguins who

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make up 30% of the total
number of penguins at the zoo.

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How many penguins
live at the zoo?

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So let's let x equal the
total number of penguins

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who live at the zoo.

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So they're telling
us that 30% of that

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are the emperor
penguins, which are 15.

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So they're saying if we
take 30%, so 30% of x

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is equal to 15.

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Or another way of saying
this is we could say,

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Instead of writing 30%, we
could write that as a decimal,

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as 0.30 of x.

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So it's 0.30 times
x is equal to 15.

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And now to solve this, we
just have to divide both sides

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by 0.30.

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So let's do that.

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0.30, 0.30, we get x is
equal to 15 divided by 0.30.

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We're going to have to do a
little arithmetic to solve

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this one.

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So if we take 0-- it's
15 divided by 0.30.

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So just as a refresher
of dividing decimals,

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we could essentially multiply
both of these by 100.

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And when you multiply
both of them by 100,

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you can't just
multiply one of them.

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Then you would get
a different answer.

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But if you multiply
both of them by 100,

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you move this decimal point over
two spots, so it becomes a 30.

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And you move this decimal over
two spots, so it becomes 1,500.

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So it really boils down
to how many times--

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the same number of times that
30 goes into 1,500 is the number

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of times that 0.30
will go into 15.

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So let's think about this.

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Let me rewrite it here,
just so it's neat.

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30 goes into 1,500, all we did
is we moved both decimals over

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to the right twice.

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Or you could say
we just multiplied

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the numerator and
the denominator

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here by 100, which
wouldn't change

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the value of the fraction.

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So 30 goes into-- let's
see, it doesn't go into 1.

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It doesn't go into 15.

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It does go into 150.

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It goes into 150 five times.

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5 times 30 is 150.

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Subtract, you get 0.

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And then 30 goes
into 0 zero times.

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So this 30 goes into
1,500 fifty times.

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So this right over
here is equal to 50.

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And you can verify it.

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Multiply 0.3 times 50,
and you will get 15.

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Now, there is another
way of doing it.

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You could say the fraction
of emperor penguins

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over the total number
of penguins at the zoo.

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So you could say 15.

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You could say 15 over the
total number of penguins

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at the zoo is equal to 30%.

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Percent literally
means 30 hundreds.

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So it's equal to
30 for every 100.

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It's equal to 30 for every 100.

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And then you could solve
for this right over here.

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And a really simple way
of doing this is you

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say, look, to go
from 30 to 15, you

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have to divide by 2 if
you're going from the right

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to the left.

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So let's divide by
2 right over here.

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So x would be equal to 50.

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15/50 is the same
thing as 30/100.

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So the total number
of penguins are 50.


Percent word problem example 2

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Language: en

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In the United States, 13 out
of every 20 cans are recycled.

00:00:06.110 --> 00:00:09.330
What percent of
cans are recycled?

00:00:09.330 --> 00:00:14.305
So 13 out of every
20 are recycled.

00:00:14.305 --> 00:00:17.625
So 13/20, or 13
over 20, could also

00:00:17.625 --> 00:00:27.930
be viewed as 13 divided
20, or 13 divided by 20.

00:00:27.930 --> 00:00:29.679
And if we do this,
we'll get a decimal,

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and it's fairly straightforward
to convert that decimal

00:00:31.970 --> 00:00:33.670
into a percentage.

00:00:33.670 --> 00:00:36.060
So 13 divided 20.

00:00:36.060 --> 00:00:38.260
We have the smaller
number in this case being

00:00:38.260 --> 00:00:40.050
divided by the larger number.

00:00:40.050 --> 00:00:42.805
So we're going to get
a value less than 1.

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Since we're going to
get a value less than 1,

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let's put a decimal
right over here.

00:00:46.580 --> 00:00:48.250
And let's add a
couple of zeroes,

00:00:48.250 --> 00:00:49.970
as many zeroes as we would need.

00:00:49.970 --> 00:00:53.590
And we could say, hey, look,
20 goes into 13 zero times.

00:00:53.590 --> 00:00:54.980
0 times 20 is 0.

00:00:54.980 --> 00:00:58.340
And then 13 minus 0 is 13.

00:00:58.340 --> 00:01:00.030
Now you bring down a 0.

00:01:00.030 --> 00:01:03.660
20 goes into 130.

00:01:03.660 --> 00:01:06.280
Let's see, it goes
in-- 5 times 20 is 100.

00:01:06.280 --> 00:01:07.730
So 6 times 20 is 120.

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So it's going to six times.

00:01:09.430 --> 00:01:12.950
6 times 20 is 120.

00:01:12.950 --> 00:01:14.280
You subtract.

00:01:14.280 --> 00:01:15.590
You get a 10.

00:01:15.590 --> 00:01:17.600
Let's bring down another 0.

00:01:17.600 --> 00:01:20.780
20 goes into 100 five times.

00:01:20.780 --> 00:01:23.830
5 times 20 is 100.

00:01:23.830 --> 00:01:26.140
And we are done.

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So this, written as
a decimal, is 0.65.

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So as a decimal, it's 0.65.

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And if you want to
write it as a percentage

00:01:37.910 --> 00:01:39.920
you essentially multiply
this number by 100.

00:01:39.920 --> 00:01:42.490
Or another way you could say
is you shift the decimal point

00:01:42.490 --> 00:01:44.540
over two spots to the right.

00:01:44.540 --> 00:01:46.930
So this is going
to be equal to 65%.

00:01:49.990 --> 00:01:52.890
Now, there's another way
you could have done it.

00:01:52.890 --> 00:01:55.240
You could have said,
look, percent literally

00:01:55.240 --> 00:01:56.770
means per hundred.

00:01:56.770 --> 00:02:08.060
So 13 out of 20 is going to
be equal to what over 100?

00:02:08.060 --> 00:02:11.970
Well, to go from 20 to a
100-- forget the denominator.

00:02:11.970 --> 00:02:14.870
To go from 20 to 100,
you would multiply by 5.

00:02:14.870 --> 00:02:17.880
So let's multiply the
numerator by 5 as well.

00:02:17.880 --> 00:02:23.360
And 13 times 5, let's see,
that's 15 plus 50, which is 65.

00:02:23.360 --> 00:02:25.519
So this would have
been a faster way

00:02:25.519 --> 00:02:27.060
to do it, especially
if you recognize

00:02:27.060 --> 00:02:29.100
it's pretty easy to
go from 20 to 100.

00:02:29.100 --> 00:02:30.420
You multiply it by 5.

00:02:30.420 --> 00:02:32.575
So we would do the
same thing with the 13.

00:02:32.575 --> 00:02:34.580
And so you would
get 65/100, which

00:02:34.580 --> 00:02:37.572
is the same thing
as 65 per-- let

00:02:37.572 --> 00:02:40.290
me write this
percent symbol-- 65%.

00:02:40.290 --> 00:02:42.450
And just a reminder,
percent literally

00:02:42.450 --> 00:02:47.080
means per hundred,
65 per hundred, 65%.


Positive and negative intervals examples

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.850
A function, f of x,
is plotted below.

00:00:02.850 --> 00:00:06.720
Highlight an interval where
f of x is less than 0.

00:00:06.720 --> 00:00:09.200
So f of x-- which is
really being plotted

00:00:09.200 --> 00:00:11.920
on the vertical axis
right over here-- x

00:00:11.920 --> 00:00:13.900
is the horizontal axis.

00:00:13.900 --> 00:00:16.329
f of x being less
than 0 really means

00:00:16.329 --> 00:00:19.130
that the graph is
below the x-axis.

00:00:19.130 --> 00:00:23.920
So the function is negative in
this interval right over here

00:00:23.920 --> 00:00:25.100
and this interval over here.

00:00:25.100 --> 00:00:27.170
So I could put this
anywhere right over here,

00:00:27.170 --> 00:00:29.290
or I could stick it
anywhere right over here.

00:00:29.290 --> 00:00:30.790
Let me stick it right over here.

00:00:30.790 --> 00:00:31.479
There we go.

00:00:31.479 --> 00:00:32.020
Got it right.

00:00:32.020 --> 00:00:33.440
Let's do a couple more.

00:00:33.440 --> 00:00:35.380
So the function
is plotted below.

00:00:35.380 --> 00:00:38.080
Highlight an interval where
f of x is greater than 0.

00:00:38.080 --> 00:00:40.170
So I could do this
area right over here

00:00:40.170 --> 00:00:42.660
where the function
is above the x-axis,

00:00:42.660 --> 00:00:44.580
or I could do this
area right over here

00:00:44.580 --> 00:00:46.930
where the function goes
way above the x-axis.

00:00:46.930 --> 00:00:49.080
Well, it even goes off the page.

00:00:49.080 --> 00:00:52.030
So let's stick it
right over here.

00:00:52.030 --> 00:00:53.930
Let's do one more.

00:00:53.930 --> 00:00:57.070
So highlight an interval where
f of x is greater than 0.

00:00:57.070 --> 00:00:59.440
Once again, I can do this
region right over here

00:00:59.440 --> 00:01:02.230
where the function
is above the x-axis,

00:01:02.230 --> 00:01:05.020
or over here where
it's above the x-axis.

00:01:05.020 --> 00:01:07.410
I'll do it here just for fun.

00:01:07.410 --> 00:01:09.205
There we go.


Evaluating composite functions example

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.870
We are told that h
of x is equal to 3x.

00:00:03.870 --> 00:00:08.386
g of t is equal to negative
2t minus 2 minus h of t.

00:00:08.386 --> 00:00:12.620
f of n is equal to negative
5n squared plus h of n.

00:00:12.620 --> 00:00:14.720
So we have three
function definitions,

00:00:14.720 --> 00:00:16.340
and two of these
function definitions

00:00:16.340 --> 00:00:18.470
are actually defined in
terms of another function--

00:00:18.470 --> 00:00:21.490
in particular, in terms
of the function h.

00:00:21.490 --> 00:00:25.740
And then we're asked to
calculate, what is h of g of 8?

00:00:25.740 --> 00:00:27.610
And this can be very
daunting, but we just

00:00:27.610 --> 00:00:30.050
have to remember that all a
function is is something that

00:00:30.050 --> 00:00:33.120
takes an input and gives
you an output for it.

00:00:33.120 --> 00:00:36.830
And this somewhat
convoluted looking statement

00:00:36.830 --> 00:00:38.830
is another way of
saying, look, we

00:00:38.830 --> 00:00:42.350
are going to take
the number 8, and we

00:00:42.350 --> 00:00:44.840
are going to input it
into the function g.

00:00:48.490 --> 00:00:51.055
And then that is going
to produce g of 8.

00:00:55.271 --> 00:00:57.270
And then we're going to
take whatever value that

00:00:57.270 --> 00:01:00.444
is and input that
into the function h.

00:01:00.444 --> 00:01:02.110
So we're going to
take this whole thing,

00:01:02.110 --> 00:01:05.840
and then we're going to input
that into the function h.

00:01:05.840 --> 00:01:09.840
And what we will have will
be h of what we inputted,

00:01:09.840 --> 00:01:15.620
h of g of 8.

00:01:15.620 --> 00:01:17.520
So let's just do it
one step at a time.

00:01:17.520 --> 00:01:20.240
Let's figure out what g of 8 is.

00:01:24.005 --> 00:01:26.630
And I'm going to color code it,
so we can keep track of things.

00:01:26.630 --> 00:01:31.540
g of 8 is equal to-- well, g of
t, we have our definition here.

00:01:31.540 --> 00:01:36.180
So our input now, 8 is going
to be our t, so our input is 8.

00:01:36.180 --> 00:01:39.200
So every place where we see a
t in this function definition,

00:01:39.200 --> 00:01:40.465
we replace it with an 8.

00:01:40.465 --> 00:01:50.911
So it's going to be negative
2 times 8 minus 2 minus--

00:01:50.911 --> 00:01:52.410
and this might be
a little daunting,

00:01:52.410 --> 00:01:54.090
but let's just replace
this t with an 8

00:01:54.090 --> 00:01:55.756
and then see if we
can make sense of it.

00:01:55.756 --> 00:01:58.540
h minus-- and let me do
it in the right color--

00:01:58.540 --> 00:02:06.900
minus 2 minus h of 8.

00:02:06.900 --> 00:02:10.430
Notice, to evaluate g of 8,
all we did is, everywhere

00:02:10.430 --> 00:02:13.970
we saw a t, we replaced
it with the input 8.

00:02:13.970 --> 00:02:15.785
Now let's see if we
can calculate this.

00:02:15.785 --> 00:02:19.620
This is going to be equal
to negative 2 times 8

00:02:19.620 --> 00:02:21.240
is negative 16.

00:02:21.240 --> 00:02:23.090
Minus 2 is negative 18.

00:02:23.090 --> 00:02:24.560
Let me do that the same.

00:02:24.560 --> 00:02:29.750
So this is going to be equals
negative 18 minus-- what

00:02:29.750 --> 00:02:32.230
is h of 8 going to be equal to?

00:02:32.230 --> 00:02:33.570
So let's do that over here.

00:02:33.570 --> 00:02:37.820
So h of 8-- this thing, h of 8.

00:02:37.820 --> 00:02:39.196
Now we go to the
definition of h.

00:02:39.196 --> 00:02:40.653
Don't worry about
later we're going

00:02:40.653 --> 00:02:42.400
to input all this
business into h again.

00:02:42.400 --> 00:02:44.275
Just let's worry about
it one step at a time.

00:02:44.275 --> 00:02:45.970
We need to calculate h of 8.

00:02:45.970 --> 00:02:51.080
So h of 8 is just going to be--
well, every time we see an x,

00:02:51.080 --> 00:02:53.960
we replace it with an 8--
it's going to be 3 times 8,

00:02:53.960 --> 00:02:56.410
which is equal to 24.

00:02:56.410 --> 00:02:58.210
So this value right
over here is 24.

00:02:58.210 --> 00:03:03.900
We are subtracting it,
so we have minus 24.

00:03:03.900 --> 00:03:06.970
Negative 18 minus 24 is what?

00:03:06.970 --> 00:03:09.310
That's negative 42.

00:03:09.310 --> 00:03:11.449
So all of this
business is going to be

00:03:11.449 --> 00:03:12.740
equal to-- did I do that right?

00:03:12.740 --> 00:03:16.060
Yeah-- negative 42.

00:03:16.060 --> 00:03:17.960
So we figured out
what g of 8 is.

00:03:17.960 --> 00:03:19.320
It is negative 42.

00:03:19.320 --> 00:03:21.990
So this right over
here is negative 42.

00:03:21.990 --> 00:03:25.130
And now we can input
negative 42 into h.

00:03:28.230 --> 00:03:34.210
Let me do it right over here.
h of negative 42-- remember,

00:03:34.210 --> 00:03:36.800
negative 42 is the
same thing as g of 8.

00:03:36.800 --> 00:03:41.765
So this is h of g of 8 is the
same thing as h of negative 42.

00:03:41.765 --> 00:03:43.140
Let me do that in
the same color.

00:03:43.140 --> 00:03:51.820
This is going to be equal to
3 times negative 42, which

00:03:51.820 --> 00:03:58.760
is equal to-- this
is negative 126.

00:03:58.760 --> 00:03:59.830
And we are done.

00:03:59.830 --> 00:04:01.290
So it seemed
convoluted at first,

00:04:01.290 --> 00:04:02.790
but if you just
keep track of what's

00:04:02.790 --> 00:04:05.580
our input, what's our
output, and really just

00:04:05.580 --> 00:04:07.840
evaluate the functions,
it should hopefully

00:04:07.840 --> 00:04:10.275
be reasonably straightforward.


How to evaluate an expression that contains function notation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.970
We're asked to evaluate
negative 2 times f of negative 6

00:00:04.970 --> 00:00:06.600
plus g of 1.

00:00:06.600 --> 00:00:08.460
And they've defined,
at least graphically,

00:00:08.460 --> 00:00:10.757
f of x and g of x here below.

00:00:10.757 --> 00:00:12.340
So let's see how we
can evaluate this.

00:00:12.340 --> 00:00:14.270
Well, to do this, we
first have to figure out

00:00:14.270 --> 00:00:16.540
what f of negative 6 is.

00:00:16.540 --> 00:00:19.760
So our input into our
function is negative 6.

00:00:19.760 --> 00:00:22.420
And we'll assume that's
along the horizontal axis.

00:00:22.420 --> 00:00:24.110
So our input is negative 6.

00:00:24.110 --> 00:00:28.690
And based on our function
definition, f of negative 6

00:00:28.690 --> 00:00:30.010
is 7.

00:00:30.010 --> 00:00:30.890
So this thing.

00:00:30.890 --> 00:00:35.690
Let me write this down. f
of negative 6 is equal to 7.

00:00:35.690 --> 00:00:36.970
And what is g of 1?

00:00:39.910 --> 00:00:42.890
Well, once again,
here's our input axis.

00:00:42.890 --> 00:00:45.410
And then the function
says that g of 1,

00:00:45.410 --> 00:00:48.660
which is right over
there, is negative 5.

00:00:48.660 --> 00:00:51.920
g of 1 is equal to negative 5.

00:00:51.920 --> 00:00:57.240
So this statement simplifies
to negative 2 times

00:00:57.240 --> 00:00:59.700
f of negative 6, which is 7.

00:00:59.700 --> 00:01:07.265
So times 7 plus g of
1, which is negative 5.

00:01:07.265 --> 00:01:12.580
So plus negative 5, which
simplifies to-- let's see.

00:01:12.580 --> 00:01:17.900
Negative 2 times 7 is
negative 14 plus negative 5,

00:01:17.900 --> 00:01:20.200
which is negative 19.

00:01:20.200 --> 00:01:22.660
And we are done.


What is a function?

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.860
A function-- and I'm
going to speak about it

00:00:02.860 --> 00:00:06.350
in very abstract terms right
now-- is something that

00:00:06.350 --> 00:00:11.134
will take an input, and
it'll munch on that input,

00:00:11.134 --> 00:00:13.050
it'll look at that input,
it will do something

00:00:13.050 --> 00:00:14.010
to that input.

00:00:14.010 --> 00:00:20.230
And based on what that input is,
it will produce a given output.

00:00:20.230 --> 00:00:22.200
What is an example
of a function?

00:00:22.200 --> 00:00:27.940
I could have something
like f of x-- and x

00:00:27.940 --> 00:00:30.200
tends to be the variable
most used for an input

00:00:30.200 --> 00:00:31.200
into the function.

00:00:31.200 --> 00:00:34.100
And the name of a
function, f tends

00:00:34.100 --> 00:00:35.650
to be the most-used variable.

00:00:35.650 --> 00:00:41.680
But we'll see that you can
use others-- is equal to,

00:00:41.680 --> 00:00:48.360
let's say, x squared,
if x is even.

00:00:48.360 --> 00:00:59.680
And let's say it is equal
to x plus 5, if x is odd.

00:00:59.680 --> 00:01:03.960
What would happen if we
input 2 into this function?

00:01:03.960 --> 00:01:05.910
The way that we would
denote inputting 2

00:01:05.910 --> 00:01:10.730
is that we would want
to evaluate f of 2.

00:01:10.730 --> 00:01:14.760
This is saying, let's input
2 into our function f.

00:01:14.760 --> 00:01:16.660
And everywhere we
see this x here,

00:01:16.660 --> 00:01:19.320
this variable-- you can kind
of use as a placeholder-- let's

00:01:19.320 --> 00:01:21.390
replace it with our input.

00:01:21.390 --> 00:01:21.980
So let's see.

00:01:21.980 --> 00:01:24.810
If 2 is even, do 2 squared.

00:01:24.810 --> 00:01:27.870
If 2 is odd, do 2 plus 5.

00:01:27.870 --> 00:01:31.670
Well, 2 is even, so we're
going to do 2 squared.

00:01:31.670 --> 00:01:36.250
In this case, f of 2 is
going to be 2 squared, or 4.

00:01:36.250 --> 00:01:40.540
Now what would f of 3 be?

00:01:40.540 --> 00:01:42.640
Well, once again, everywhere
we see this variable,

00:01:42.640 --> 00:01:44.135
we'll replace it with our input.

00:01:48.250 --> 00:01:54.950
So f of 3, 3 squared if 3 is
even, 3 plus 5 if 3 is odd.

00:01:54.950 --> 00:01:57.390
Well, 3 is odd, so it's
going to be 3 plus 5.

00:01:57.390 --> 00:02:02.670
It is going to be equal to 8.

00:02:02.670 --> 00:02:04.200
You might say, OK,
that's neat, Sal.

00:02:04.200 --> 00:02:05.790
This was kind of
an interesting way

00:02:05.790 --> 00:02:09.139
to define a function, a way to
kind of munch on these numbers.

00:02:09.139 --> 00:02:12.217
But I could have done this
with traditional equations

00:02:12.217 --> 00:02:13.800
in some way, especially
if you allowed

00:02:13.800 --> 00:02:16.140
me to use the squirrelly
bracket thing.

00:02:16.140 --> 00:02:19.710
What can a function do that
maybe my traditional toolkits

00:02:19.710 --> 00:02:22.160
might have not been
as expressive about?

00:02:22.160 --> 00:02:25.880
Well, you could even do
a function like this.

00:02:25.880 --> 00:02:27.790
Let me not use f
and x anymore, just

00:02:27.790 --> 00:02:30.800
to show you that the notation
is more general than that.

00:02:30.800 --> 00:02:47.050
I could say h of a is equal to
the next largest number that

00:02:47.050 --> 00:02:57.886
starts with the same
letter as variable a.

00:03:00.460 --> 00:03:03.330
And we're going to assume
that we're dealing in English.

00:03:03.330 --> 00:03:09.930
Given that, what is
h of 2 going to be?

00:03:09.930 --> 00:03:12.890
Well, 2 starts with a T.
What's the next largest

00:03:12.890 --> 00:03:14.210
number that starts with a T?

00:03:14.210 --> 00:03:18.610
Well, it's going
to be equal to 3.

00:03:18.610 --> 00:03:25.210
Now what would h of-- I don't
know, let's think about this,

00:03:25.210 --> 00:03:27.990
h of 8 be equal to?

00:03:27.990 --> 00:03:31.570
Well, 8 starts with an E.
The next largest number

00:03:31.570 --> 00:03:37.400
that starts with an E-- it's
not 9, 10-- it would be 11.

00:03:37.400 --> 00:03:42.100
And so now you see it's a
very, very, very general tool.

00:03:42.100 --> 00:03:44.630
This h function that we just
defined, we'll look at it.

00:03:44.630 --> 00:03:46.500
We'll look at the
letter that the number

00:03:46.500 --> 00:03:47.869
starts with in English.

00:03:47.869 --> 00:03:49.910
So it's doing this really,
really, really, really

00:03:49.910 --> 00:03:50.990
wacky thing.

00:03:50.990 --> 00:03:53.070
Now not all functions
have to be this wacky.

00:03:53.070 --> 00:03:55.950
In fact, you have already
been dealing with functions.

00:03:55.950 --> 00:04:02.340
You have seen things like
y is equal to x plus 1.

00:04:02.340 --> 00:04:04.080
This can be viewed
as a function.

00:04:04.080 --> 00:04:07.520
We could write this
as y is a function

00:04:07.520 --> 00:04:11.760
of x, which is
equal to x plus 1.

00:04:11.760 --> 00:04:16.390
If you give as an
input-- let me write

00:04:16.390 --> 00:04:23.650
it this way-- for example,
when x is 0 we could say f of 0

00:04:23.650 --> 00:04:25.710
is equal to, well, you take 0.

00:04:25.710 --> 00:04:26.830
You add 1.

00:04:26.830 --> 00:04:28.440
It's equal to 1.

00:04:28.440 --> 00:04:30.447
f of 2 is equal to 2.

00:04:30.447 --> 00:04:31.780
You've already done this before.

00:04:31.780 --> 00:04:33.760
You've done things
where you said, look,

00:04:33.760 --> 00:04:37.250
let me make a table of
x and put our y's there.

00:04:37.250 --> 00:04:40.230
When x is 0, y is 1.

00:04:40.230 --> 00:04:42.990
I'm sorry.

00:04:42.990 --> 00:04:44.150
I made a little mistake.

00:04:44.150 --> 00:04:46.660
Where f of 2 is equal to 3.

00:04:46.660 --> 00:04:48.920
And you've done this before
with tables where you say,

00:04:48.920 --> 00:04:51.470
look, x and y.

00:04:51.470 --> 00:04:53.650
When x is 0, y is 1.

00:04:53.650 --> 00:04:57.151
When x is 2, y is 3.

00:04:57.151 --> 00:04:59.400
You might say, well, what
was the whole point of using

00:04:59.400 --> 00:05:01.310
the function
notation here to say

00:05:01.310 --> 00:05:03.820
f of x is equal to x plus 1?

00:05:03.820 --> 00:05:07.390
The whole point is to think
in these more general terms.

00:05:07.390 --> 00:05:09.140
For something like
this, you didn't really

00:05:09.140 --> 00:05:11.417
have to introduce
function notations.

00:05:11.417 --> 00:05:14.000
But it doesn't hurt to introduce
function notations because it

00:05:14.000 --> 00:05:17.820
makes it very clear that
the function takes an input,

00:05:17.820 --> 00:05:21.520
takes my x-- in this
definition it munches on it.

00:05:21.520 --> 00:05:23.500
It says, OK, x plus 1.

00:05:23.500 --> 00:05:26.080
And then it produces
1 more than it.

00:05:26.080 --> 00:05:29.460
So here, whatever the input
is, the output is 1 more

00:05:29.460 --> 00:05:31.100
than that original function.

00:05:31.100 --> 00:05:32.350
Now I know what you're asking.

00:05:32.350 --> 00:05:32.850
All right.

00:05:32.850 --> 00:05:35.210
Well, what is not
a function then?

00:05:35.210 --> 00:05:36.750
Well, remember,
we said a function

00:05:36.750 --> 00:05:39.850
is something that takes an
input and produces only one

00:05:39.850 --> 00:05:43.130
possible output for
that given input.

00:05:43.130 --> 00:05:46.580
For example-- and let
me look at a visual way

00:05:46.580 --> 00:05:49.100
of thinking about a function
this time, or a relationship,

00:05:49.100 --> 00:05:53.190
I should say-- let's
say that's our y-axis,

00:05:53.190 --> 00:05:55.463
and this right over
here is our x-axis.

00:05:58.241 --> 00:06:05.720
Let me draw a circle
here that has radius 2.

00:06:05.720 --> 00:06:07.950
So it's a circle of radius 2.

00:06:07.950 --> 00:06:09.300
This is negative 2.

00:06:09.300 --> 00:06:11.120
This is positive 2.

00:06:11.120 --> 00:06:12.730
This is negative 2.

00:06:12.730 --> 00:06:15.940
So my circle, it's
centered at the origin.

00:06:15.940 --> 00:06:17.330
It has radius 2.

00:06:17.330 --> 00:06:20.560
That's my best attempt
at drawing the circle.

00:06:20.560 --> 00:06:21.920
Let me fill it in.

00:06:21.920 --> 00:06:23.100
So this is a circle.

00:06:23.100 --> 00:06:25.070
The equation of
this circle is going

00:06:25.070 --> 00:06:28.370
to be x squared plus y squared
is equal to the radius squared,

00:06:28.370 --> 00:06:31.360
is equal to 2 squared,
or it's equal to 4.

00:06:31.360 --> 00:06:34.027
The question is, is this
relationship between x and y--

00:06:34.027 --> 00:06:35.610
here I've expressed
it as an equation.

00:06:35.610 --> 00:06:40.020
Here I've visually drawn all
of the x's and y's that satisfy

00:06:40.020 --> 00:06:43.220
this equation-- is this
relationship between x

00:06:43.220 --> 00:06:45.280
and y a function?

00:06:45.280 --> 00:06:47.060
And we can see
visually that it's not

00:06:47.060 --> 00:06:48.280
going to be a function.

00:06:48.280 --> 00:06:49.850
You pick a given x.

00:06:49.850 --> 00:06:53.820
Let's say x is equal to 1.

00:06:53.820 --> 00:06:57.150
There's two possible y's
that are associated with it,

00:06:57.150 --> 00:07:00.237
this y up here and
this y down here.

00:07:00.237 --> 00:07:02.570
We could even solve for that
by looking at the equation.

00:07:02.570 --> 00:07:06.210
When x is equal to 1, we
get 1 squared plus y squared

00:07:06.210 --> 00:07:07.073
is equal to 4.

00:07:07.073 --> 00:07:11.390
1 plus y squared is equal to 4.

00:07:11.390 --> 00:07:15.720
Or subtracting 1 from both
sides, y squared is equal to 3.

00:07:15.720 --> 00:07:20.830
Or y is equal to the positive or
the negative square root of 3.

00:07:20.830 --> 00:07:23.720
This right over here is the
positive square root of 3,

00:07:23.720 --> 00:07:26.700
and this right over here is
the negative square root of 3.

00:07:26.700 --> 00:07:28.810
So this situation,
this relationship

00:07:28.810 --> 00:07:34.130
where I inputted a 1
into my little box here,

00:07:34.130 --> 00:07:36.680
and associated with
the 1, I associate

00:07:36.680 --> 00:07:40.510
both a positive square root of
3 and a negative square root

00:07:40.510 --> 00:07:47.000
of 3, this is not a function.

00:07:47.000 --> 00:07:51.540
I cannot associate with my
input two different outputs.

00:07:51.540 --> 00:07:56.240
I can only have one
output for a given input.


Multi-step word problem with fractions and units of time

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.750
Jess recently started a business
producing cosmetic products

00:00:04.750 --> 00:00:06.950
made from natural ingredients.

00:00:06.950 --> 00:00:09.450
She wants to
advertise her products

00:00:09.450 --> 00:00:12.870
by distributing bags of
samples in her neighborhood.

00:00:12.870 --> 00:00:15.360
She calculated that it
takes her two minutes

00:00:15.360 --> 00:00:17.970
to prepare each bag.

00:00:17.970 --> 00:00:21.640
How many hours--
they give us minutes

00:00:21.640 --> 00:00:24.070
here, now they want
hours-- how many hours will

00:00:24.070 --> 00:00:27.650
it take to prepare
1,200 bags of samples

00:00:27.650 --> 00:00:33.120
if she asks seven friends
to help her with her work?

00:00:33.120 --> 00:00:35.050
So the one thing we
could think about is,

00:00:35.050 --> 00:00:38.280
how long would it take
her to do it by herself?

00:00:38.280 --> 00:00:44.510
So it takes her, by herself,
two minutes per bag.

00:00:44.510 --> 00:00:54.540
And then we have 1,200 bags,
not 12 bags, 1,200 bags

00:00:54.540 --> 00:00:57.850
which means it's going to
take her two minutes for each

00:00:57.850 --> 00:00:58.900
of the 1,200 bags.

00:00:58.900 --> 00:01:04.294
Well that's 2 times
1,200 is 2,400 minutes.

00:01:04.294 --> 00:01:06.460
Now that's if she was just
working on it by herself,

00:01:06.460 --> 00:01:08.330
but she's got seven friends.

00:01:08.330 --> 00:01:10.140
And we can assume
that they would all

00:01:10.140 --> 00:01:12.689
take about two minutes
to prepare each bag.

00:01:12.689 --> 00:01:14.480
Now this is a little
bit tricky because you

00:01:14.480 --> 00:01:16.432
might want to divide
this by 7 saying, hey,

00:01:16.432 --> 00:01:18.390
we're going to have seven
times as many people.

00:01:18.390 --> 00:01:20.780
But you remember, she
asks seven friends

00:01:20.780 --> 00:01:22.530
so there's actually
eight people involved.

00:01:22.530 --> 00:01:25.540
There's Jess and
her seven friends.

00:01:25.540 --> 00:01:27.990
So it's actually going
to be eight times faster

00:01:27.990 --> 00:01:31.022
than if Jess did it by herself.

00:01:31.022 --> 00:01:32.480
This is if she only
had one person.

00:01:32.480 --> 00:01:34.021
If you're going to
have eight people,

00:01:34.021 --> 00:01:37.670
you are going to take
eight times less time.

00:01:37.670 --> 00:01:41.150
So with eight people, it's
going to be 2,400 divided

00:01:41.150 --> 00:01:45.310
by 8 is 300 minutes.

00:01:45.310 --> 00:01:48.380
And we're almost done, except
for the fact that they asked

00:01:48.380 --> 00:01:52.030
for our answer in
terms of hours.

00:01:52.030 --> 00:01:55.450
So 300 minutes is
how many hours?

00:01:55.450 --> 00:01:58.260
Well you have 60
minutes per hour

00:01:58.260 --> 00:02:02.680
so you could say 300
minutes-- I'll just write min

00:02:02.680 --> 00:02:08.389
for short-- divided
by 60 minutes per hour

00:02:08.389 --> 00:02:14.210
gives us 300 divided
by 60 is 5 hours.

00:02:14.210 --> 00:02:17.840
So it will take Jess and her
seven friends, a total of eight

00:02:17.840 --> 00:02:23.393
people, five hours to prepare
all of her sample bags.


Multi-step word problem with addition, subtraction, and multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.740
Sandy is taking a
cab in New York City.

00:00:03.740 --> 00:00:08.230
The taxi fare includes a base
charge of $2.50 as soon as you

00:00:08.230 --> 00:00:11.870
enter the cab, plus an
additional $2.50 per mile

00:00:11.870 --> 00:00:12.990
traveled.

00:00:12.990 --> 00:00:15.480
If Sandy starts off
with a $20 bill,

00:00:15.480 --> 00:00:19.390
takes the cab for five
miles, and leaves a $1 tip,

00:00:19.390 --> 00:00:22.540
how much money will
she be left with?

00:00:22.540 --> 00:00:25.660
Well let's first just think
about how much this five mile

00:00:25.660 --> 00:00:29.000
cab ride is going to cost her.

00:00:29.000 --> 00:00:32.425
So cab-- I'll call
it the cab fare.

00:00:36.820 --> 00:00:41.860
So it's going to be $2.50 just
when you get into the cab,

00:00:41.860 --> 00:00:46.310
plus $2.50 per mile.

00:00:46.310 --> 00:00:55.050
Well she's going
to go for 5 miles--

00:00:55.050 --> 00:01:03.010
let me write it this way--
times $2.50 per miles.

00:01:03.010 --> 00:01:08.630
So the cab fare here is just
going to be $2.50, this $2.50,

00:01:08.630 --> 00:01:10.404
plus 5 times $2.50.

00:01:10.404 --> 00:01:16.970
Let's see, 5 times
$2.50, that is $12.50.

00:01:16.970 --> 00:01:22.070
So this is plus $12.50.

00:01:22.070 --> 00:01:27.280
$2.50 plus $12.50
is going to be $15.

00:01:27.280 --> 00:01:31.230
So just the cab fare
is going to be $15.

00:01:31.230 --> 00:01:34.700
And then Sandy gives a $1 tip.

00:01:34.700 --> 00:01:41.700
So fare plus tip is going
to be equal to the cab fare

00:01:41.700 --> 00:01:47.124
$15 plus $1 which
is equal to $16.

00:01:47.124 --> 00:01:49.040
So then answering the
question, how much money

00:01:49.040 --> 00:01:51.630
would she be left
with with a $20 bill?

00:01:51.630 --> 00:01:54.420
Well we could
write her change is

00:01:54.420 --> 00:01:57.010
going to be, she's going
to give a $20 bill,

00:01:57.010 --> 00:02:02.560
the fare plus tip
is $16, minus $16.

00:02:02.560 --> 00:02:07.069
So she's going to have
$4 left in change.


Multi-step word problem with decimals and subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:06.010
Joelle is working on a school
report in a word processor.

00:00:06.010 --> 00:00:10.880
She wants to place a figure
which is 13.09 centimeters wide

00:00:10.880 --> 00:00:13.550
so that it appears
centered on the page.

00:00:13.550 --> 00:00:17.200
The width of the page
is 21.59 centimeters.

00:00:17.200 --> 00:00:20.540
How much margin should
she leave on the left

00:00:20.540 --> 00:00:22.130
so that the figure is centered?

00:00:22.130 --> 00:00:24.020
So let's visualize
what's going on.

00:00:24.020 --> 00:00:28.060
We have a page that's
21.59 centimeters wide.

00:00:28.060 --> 00:00:34.160
So this distance right over
here is 21.59 centimeters.

00:00:34.160 --> 00:00:36.720
So this right over
here, that is our page.

00:00:36.720 --> 00:00:38.530
Let me draw the actual page.

00:00:38.530 --> 00:00:41.270
Let me draw that in white,
since paper tends to be white.

00:00:41.270 --> 00:00:45.510
So this is her page.

00:00:45.510 --> 00:00:49.385
It's 21.59 centimeters wide.

00:00:49.385 --> 00:00:50.380
It's just like that.

00:00:50.380 --> 00:00:52.970
It's a little unusual
dimension for a piece of paper.

00:00:52.970 --> 00:00:54.062
But we'll go with it.

00:00:54.062 --> 00:00:55.770
And then she wants to
place a figure that

00:00:55.770 --> 00:00:59.930
is 13.09 centimeters wide
in the center of the page.

00:00:59.930 --> 00:01:03.515
So the figure-- I'll
do it in magenta.

00:01:03.515 --> 00:01:05.580
That's not magenta.

00:01:05.580 --> 00:01:08.690
The figure is 13.09 centimeters.

00:01:08.690 --> 00:01:11.670
So let's say that this is
the figure right over here.

00:01:11.670 --> 00:01:14.620
So this is the
figure that she wants

00:01:14.620 --> 00:01:16.690
to place in the
middle of the page.

00:01:16.690 --> 00:01:23.880
So this is 13.09 centimeters.

00:01:23.880 --> 00:01:26.180
So the question is, how
much margin should she

00:01:26.180 --> 00:01:29.230
leave on the left so that
the figure is centered?

00:01:29.230 --> 00:01:34.560
So they're really asking,
what should this distance

00:01:34.560 --> 00:01:37.370
be here so that the
figure is centered?

00:01:37.370 --> 00:01:39.720
And the key thing is, if
the figure is centered,

00:01:39.720 --> 00:01:42.990
this distance is going to be
the same as this distance,

00:01:42.990 --> 00:01:45.060
so that the figure is
right in the middle.

00:01:45.060 --> 00:01:47.700
Whatever this is, it's going
to be the same as this.

00:01:47.700 --> 00:01:49.360
So one way we could
think about it--

00:01:49.360 --> 00:01:56.150
we could say, OK, let's take
13.09 centimeters from 21.59

00:01:56.150 --> 00:01:56.720
centimeters.

00:01:56.720 --> 00:02:00.650
And if we do that,
then we'll know

00:02:00.650 --> 00:02:04.190
how much leftover space
this plus this combined is.

00:02:04.190 --> 00:02:06.540
And then we just want to
split that space in half.

00:02:06.540 --> 00:02:08.430
We want half of it to
go on the right side,

00:02:08.430 --> 00:02:11.070
and we want half of it
to go on the left side.

00:02:11.070 --> 00:02:14.440
So let's see how much this left
margin and the right margin

00:02:14.440 --> 00:02:15.920
combined need to be.

00:02:15.920 --> 00:02:18.330
And that's just 21.59.

00:02:18.330 --> 00:02:26.410
That's just going to be
21.59 centimeters minus 13.09

00:02:26.410 --> 00:02:27.120
centimeters.

00:02:27.120 --> 00:02:29.920
And this, we get to-- let's see.

00:02:29.920 --> 00:02:38.262
This is 8.50 centimeters,
which is the combined margins.

00:02:38.262 --> 00:02:39.970
Not just the left or
not just the right--

00:02:39.970 --> 00:02:43.910
this is the left plus the right
margin, is 8.50 centimeters.

00:02:43.910 --> 00:02:46.880
This is this distance
plus this distance.

00:02:46.880 --> 00:02:49.630
Now, if we want to figure out
what the left margin should be,

00:02:49.630 --> 00:02:50.630
if we want to center it.

00:02:50.630 --> 00:02:52.470
Well, we just want
to split this evenly

00:02:52.470 --> 00:02:54.370
between the left and
the right margin.

00:02:54.370 --> 00:02:56.780
So we just want to
divide this by 2.

00:02:56.780 --> 00:03:02.870
So if you divide 18.50 by 2--
or multiply it by 1/2, I guess.

00:03:02.870 --> 00:03:05.460
That might be a little bit
more obvious-- either way.

00:03:05.460 --> 00:03:11.260
So if we multiply this by 1/2,
you are left with 1/2 of 8.50,

00:03:11.260 --> 00:03:14.810
is 4.25 centimeters.

00:03:14.810 --> 00:03:16.790
So if we want it
centered, this is

00:03:16.790 --> 00:03:19.420
going to be 4.25 centimeters.

00:03:19.420 --> 00:03:23.360
And of course, this is going to
be 4.25 centimeters, as well.

00:03:23.360 --> 00:03:26.200
And you see, when
you add 4.25 to 4.25,

00:03:26.200 --> 00:03:27.990
you get 8.5 centimeters.

00:03:27.990 --> 00:03:30.410
And then you add that to 13.09.

00:03:30.410 --> 00:03:32.730
You get the entire width
of the piece of paper.

00:03:32.730 --> 00:03:35.290
So how much margin should
she leave on the left?

00:03:35.290 --> 00:03:38.602
4.25 centimeters.


Comparing linear functions word problems 3

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.010
Charles and Tammy live the
same distance from work,

00:00:04.010 --> 00:00:07.860
and they started walking
to work at the same time.

00:00:07.860 --> 00:00:10.510
They both walked
at constant speeds,

00:00:10.510 --> 00:00:15.340
though not necessarily
at the same speed.

00:00:15.340 --> 00:00:19.190
Charles' distance from work is
shown in the following table.

00:00:19.190 --> 00:00:22.750
So at 50 seconds, he's
830 meters from work.

00:00:22.750 --> 00:00:25.590
Then another 50 seconds
goes by, and now he's

00:00:25.590 --> 00:00:27.110
760 meters from work.

00:00:27.110 --> 00:00:28.880
Another 50 seconds
goes by, and he's

00:00:28.880 --> 00:00:31.460
690 meters from work,
which makes sense.

00:00:31.460 --> 00:00:34.190
He's getting closer and closer
to work as seconds pass by.

00:00:34.190 --> 00:00:36.080
He's walking to work.

00:00:36.080 --> 00:00:38.910
Which of these sentences
could possibly be true?

00:00:38.910 --> 00:00:40.790
Select all that apply.

00:00:40.790 --> 00:00:42.790
So what are these
sentences talking about?

00:00:42.790 --> 00:00:44.831
So it looks like they're
all talking about Tammy.

00:00:44.831 --> 00:00:46.370
Tammy started 830
meters from work

00:00:46.370 --> 00:00:48.800
and walked towards work
at 2 meters per second.

00:00:48.800 --> 00:00:52.590
She started 900 meters from
work and walked towards work

00:00:52.590 --> 00:00:54.579
at 7/5 meters per second.

00:00:54.579 --> 00:00:56.120
So all of these seem
to be statements

00:00:56.120 --> 00:00:59.580
about where did Tammy start
and how quickly did she

00:00:59.580 --> 00:01:00.700
walk to work?

00:01:00.700 --> 00:01:04.180
So let's think about which of
these could possibly be true.

00:01:04.180 --> 00:01:06.290
Well, the one thing
that we know is

00:01:06.290 --> 00:01:09.300
that they live the same
distance from work.

00:01:09.300 --> 00:01:11.010
So the distance
that Charles started

00:01:11.010 --> 00:01:14.010
walking from work, that's
the same distance that Tammy

00:01:14.010 --> 00:01:15.440
started walking from work.

00:01:15.440 --> 00:01:17.780
So let's figure out
what that distance is.

00:01:17.780 --> 00:01:20.250
So we see that every
50 seconds, so if we

00:01:20.250 --> 00:01:25.810
go from 50 to 100 seconds, so
if we add 50 seconds, or 50

00:01:25.810 --> 00:01:29.030
seconds go by, it
looks like Charles

00:01:29.030 --> 00:01:32.230
traveled another-- let's see.

00:01:32.230 --> 00:01:38.130
To go from 830 to 760,
his distance from work

00:01:38.130 --> 00:01:43.985
decreased by 70 meters.

00:01:43.985 --> 00:01:45.610
And let's see if that
keeps holding up.

00:01:45.610 --> 00:01:47.920
It should because they're
moving at constant speeds.

00:01:47.920 --> 00:01:53.230
Another 50 seconds goes by,
and his distance from work

00:01:53.230 --> 00:01:56.150
decreases by another 70 meters.

00:01:56.150 --> 00:01:58.640
So what would have happened
50 seconds in the past?

00:01:58.640 --> 00:02:00.660
So what would
happen at 0 seconds?

00:02:00.660 --> 00:02:04.130
So 50 seconds before
he's at 830 meters,

00:02:04.130 --> 00:02:07.550
he would have been 70
meters further from work.

00:02:07.550 --> 00:02:11.410
So he would have been
900 meters from work,

00:02:11.410 --> 00:02:13.050
and we see that it's consistent.

00:02:13.050 --> 00:02:17.560
When 50 seconds pass by,
when we go from 0 seconds

00:02:17.560 --> 00:02:22.230
to 50 seconds, he goes from 900
meters away to 70 meters away.

00:02:22.230 --> 00:02:26.030
So once again, his distance
from work decreases by 70.

00:02:26.030 --> 00:02:28.850
Then another 50 seconds goes
by, it decreases by 70 again.

00:02:28.850 --> 00:02:31.870
Another 50 seconds, it
decreases by 70 again.

00:02:31.870 --> 00:02:36.920
So at time 0, Charles was
900 meters away from work.

00:02:36.920 --> 00:02:40.080
Now, we know that Tammy and
Charles live the same distance

00:02:40.080 --> 00:02:40.720
away from work.

00:02:40.720 --> 00:02:45.490
So Tammy would have also started
900 meters away from work.

00:02:45.490 --> 00:02:47.930
Now, they tell us that
they don't necessarily

00:02:47.930 --> 00:02:50.060
walk at the same speed.

00:02:50.060 --> 00:02:52.240
So we don't know what
Tammy's speed is.

00:02:52.240 --> 00:02:55.730
This hasn't in any way
constrained Tammy's speed.

00:02:55.730 --> 00:02:58.510
So we know that she
started 900 meters away.

00:02:58.510 --> 00:03:01.130
So we know that this
can't be the case.

00:03:01.130 --> 00:03:02.340
That says 830.

00:03:02.340 --> 00:03:03.840
We know that this
can't be the case.

00:03:03.840 --> 00:03:06.006
We know that this can't be
the case because they all

00:03:06.006 --> 00:03:08.900
talk about starting at a
distance other than 900 meters.

00:03:08.900 --> 00:03:10.400
And over here,
both of these, they

00:03:10.400 --> 00:03:12.510
say she started at
900 meters from work

00:03:12.510 --> 00:03:16.060
and walked towards work
at 7/5 meters per second.

00:03:16.060 --> 00:03:17.310
Well, that's possible.

00:03:17.310 --> 00:03:18.890
It's not necessarily true.

00:03:18.890 --> 00:03:21.120
They just haven't given us
information on her speed.

00:03:21.120 --> 00:03:23.500
But given the
information so far,

00:03:23.500 --> 00:03:25.880
this is definitely
a possibility.

00:03:25.880 --> 00:03:28.360
Likewise, Tammy
started 900 meters

00:03:28.360 --> 00:03:29.850
from work-- we
know that is true--

00:03:29.850 --> 00:03:33.000
and walked towards work
at 5/7 meters per second.

00:03:33.000 --> 00:03:35.070
Well, once again,
that's a possibility.

00:03:35.070 --> 00:03:39.120
That hasn't been ruled
out by the information

00:03:39.120 --> 00:03:40.160
right over here.

00:03:40.160 --> 00:03:45.197
So these two are
both possibly true.


Comparing linear functions word problems 1

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.860
Nick and Alyssa are
racing up a wall.

00:00:03.860 --> 00:00:06.750
Alyssa's height on the wall
is given by the equation.

00:00:06.750 --> 00:00:09.630
a is equal to 1/3
t plus 5-- so it

00:00:09.630 --> 00:00:12.100
sounds like they're wall
climbers of some kind-- where

00:00:12.100 --> 00:00:15.720
a is Alyssa's height in feet
after climbing 4t seconds.

00:00:15.720 --> 00:00:18.390
Nick started racing at
the same time as Alyssa

00:00:18.390 --> 00:00:21.090
and is also climbing
at a constant speed.

00:00:21.090 --> 00:00:22.980
His height is shown in
the following table.

00:00:22.980 --> 00:00:25.030
So this is t in seconds,
time in seconds.

00:00:25.030 --> 00:00:26.570
This is height in feet.

00:00:26.570 --> 00:00:29.842
Who started out
higher, Alyssa or Nick?

00:00:29.842 --> 00:00:31.550
So to figure out their
starting position,

00:00:31.550 --> 00:00:33.008
we just need to
figure out what was

00:00:33.008 --> 00:00:34.370
their height at time equals 0.

00:00:34.370 --> 00:00:37.099
That's when this
whole race started.

00:00:37.099 --> 00:00:38.890
For Alyssa, this is
pretty straightforward.

00:00:38.890 --> 00:00:43.330
When time is equal to 0,
you have 1/3 times 0 plus 5.

00:00:43.330 --> 00:00:45.110
Well, that's just
going to be 5 feet.

00:00:45.110 --> 00:00:48.590
So Alyssa's starting
position is at 5 feet

00:00:48.590 --> 00:00:51.580
when time is equal to 0.

00:00:51.580 --> 00:00:54.240
Now, let's think about Nick's
height at time equals 0,

00:00:54.240 --> 00:00:55.740
and there's a couple
of ways that we

00:00:55.740 --> 00:00:57.410
can go about doing this.

00:00:57.410 --> 00:01:01.390
One is just to back up, to kind
go backwards on this chart.

00:01:01.390 --> 00:01:03.140
So let me show you
what I'm talking about.

00:01:03.140 --> 00:01:07.150
So if this is time, and this is,
let's say, n for Nick's height

00:01:07.150 --> 00:01:09.220
because we have a
for Alyssa's height.

00:01:09.220 --> 00:01:11.440
So I'm going to make
a little table here.

00:01:11.440 --> 00:01:16.110
We already know that at time
6 seconds, or after 6 seconds,

00:01:16.110 --> 00:01:19.460
he's 6 feet in the
air or along the wall.

00:01:19.460 --> 00:01:24.780
At time 8, he is 7 feet in
the air or along the wall.

00:01:24.780 --> 00:01:28.740
And at time 10, he's gotten
to a height of 8 feet.

00:01:28.740 --> 00:01:31.650
So what's happening here?

00:01:31.650 --> 00:01:40.440
Every time 2 seconds goes
by, he increases in 1 foot.

00:01:40.440 --> 00:01:44.820
You have another 2 seconds, he
increases in height by 1 foot.

00:01:44.820 --> 00:01:45.950
So you could go backwards.

00:01:45.950 --> 00:01:48.590
If we take away 2
seconds to 4 seconds,

00:01:48.590 --> 00:01:50.880
he will decrease in
height by 1 foot.

00:01:50.880 --> 00:01:53.450
If we go back
another 2 seconds, he

00:01:53.450 --> 00:01:55.680
will decrease in height
by another 1 foot.

00:01:55.680 --> 00:01:57.080
The reason why we
can say this is

00:01:57.080 --> 00:01:59.680
because we know he's
climbing at a constant speed.

00:01:59.680 --> 00:02:03.430
So if we decrease by another 2
seconds to our starting time,

00:02:03.430 --> 00:02:06.140
then we know that he would
have been 1 foot lower,

00:02:06.140 --> 00:02:08.169
so he would have been at 3 feet.

00:02:08.169 --> 00:02:11.100
So just like that, we now
know at time equals 0,

00:02:11.100 --> 00:02:14.170
Nick's height is
3 feet in the air.

00:02:14.170 --> 00:02:16.600
So Alyssa started
out higher than Nick.

00:02:16.600 --> 00:02:19.840
So this right over here
would be the correct answer.

00:02:19.840 --> 00:02:22.490
Now, the other way to do it is
set up an equation just like we

00:02:22.490 --> 00:02:25.520
had for Alyssa and
substitute for time equals 0.

00:02:25.520 --> 00:02:27.960
And the way to do
that is to recognize

00:02:27.960 --> 00:02:30.350
that Nick's height,
as a function of time,

00:02:30.350 --> 00:02:32.220
is also going to be
a linear equation.

00:02:32.220 --> 00:02:34.382
Because they're both climbing.

00:02:34.382 --> 00:02:36.090
They're both climbing
at constant speeds,

00:02:36.090 --> 00:02:38.810
or we know that Nick is
climbing at a constant speed.

00:02:38.810 --> 00:02:40.784
So Nick's height, as
a function of time,

00:02:40.784 --> 00:02:42.700
is going to look like--
Nick's height is going

00:02:42.700 --> 00:02:46.120
to be some slope,
some rate of change,

00:02:46.120 --> 00:02:50.180
essentially his height
per second at times

00:02:50.180 --> 00:02:55.500
time plus his initial position.

00:02:55.500 --> 00:02:58.670
So how can we solve
for m, the slope,

00:02:58.670 --> 00:03:00.830
and his initial position?

00:03:00.830 --> 00:03:04.580
Well, the slope is just his
rate of change of height,

00:03:04.580 --> 00:03:06.810
so it's literally how
much does his height

00:03:06.810 --> 00:03:08.900
change per unit time?

00:03:08.900 --> 00:03:14.610
So m right over here, m is just
going to be for a unit time,

00:03:14.610 --> 00:03:18.810
for change in time, how
much is his height changing?

00:03:18.810 --> 00:03:22.020
And his height is--
we used the letter n.

00:03:22.020 --> 00:03:28.200
So we already know that
when time increases by 2,

00:03:28.200 --> 00:03:33.590
his height increases by 1 foot.

00:03:33.590 --> 00:03:36.100
So we know that m
is equal to 1/2.

00:03:36.100 --> 00:03:38.629
He increases 1/2
feet per second.

00:03:38.629 --> 00:03:40.420
And you see that there
because it takes him

00:03:40.420 --> 00:03:42.650
2 seconds to go 1 foot.

00:03:42.650 --> 00:03:44.110
So we can fill in m here.

00:03:44.110 --> 00:03:55.320
So we know now that n is
equal to 1/2 t plus b.

00:03:55.320 --> 00:03:57.350
Now, to solve for
b, you could just

00:03:57.350 --> 00:03:58.740
substitute one of these points.

00:03:58.740 --> 00:04:02.470
All of these points must satisfy
this equation right over here.

00:04:02.470 --> 00:04:04.600
So we could use the point 6.

00:04:04.600 --> 00:04:08.920
So if we put a 6 in
here, so when time is 6,

00:04:08.920 --> 00:04:11.240
we know that n is 6.

00:04:11.240 --> 00:04:14.490
So you have 6 is equal
to 1/2 times 6 plus b.

00:04:14.490 --> 00:04:17.950
Or you get 6 is
equal to 3 plus b.

00:04:17.950 --> 00:04:23.560
Subtract 3 from both sides,
you get b is equal to 3.

00:04:23.560 --> 00:04:24.450
So there you have it.

00:04:24.450 --> 00:04:27.130
You get Nick's equation,
or Nick's height

00:04:27.130 --> 00:04:28.210
as a function of time.

00:04:28.210 --> 00:04:29.880
Nick as a function
of time is going

00:04:29.880 --> 00:04:33.840
to be equal to 1/2 t plus 3.

00:04:33.840 --> 00:04:36.280
So now we have an equation
just like Alyssa's.

00:04:36.280 --> 00:04:38.970
And we can say, well,
when time is equal to 0,

00:04:38.970 --> 00:04:40.980
he's at a height of
3, which is lower

00:04:40.980 --> 00:04:43.200
than Alyssa's initial height.


Comparing linear functions 3

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.510
f is a linear function whose
table of values is shown below.

00:00:04.510 --> 00:00:06.540
And they give us three
different x-values

00:00:06.540 --> 00:00:08.980
and the corresponding
f of x values.

00:00:08.980 --> 00:00:13.210
Which graphs show functions that
are increasing faster than f?

00:00:13.210 --> 00:00:15.720
So when we're talking
about increasing faster,

00:00:15.720 --> 00:00:18.090
we're really talking
about a higher rate

00:00:18.090 --> 00:00:22.180
of change of y with respect to
f, or a higher rate of change

00:00:22.180 --> 00:00:24.380
of the vertical
axis with respect

00:00:24.380 --> 00:00:27.450
to the horizontal axis, which
is another way of saying which

00:00:27.450 --> 00:00:32.890
of these have a steeper
slope than the function f?

00:00:32.890 --> 00:00:36.150
So let's see what the
change in our vertical axis

00:00:36.150 --> 00:00:40.270
is with respect to our change
in our horizontal axis.

00:00:40.270 --> 00:00:44.550
Once again, the Greek
letter-- this triangle

00:00:44.550 --> 00:00:47.790
is the Greek letter delta, which
is shorthand for "change in."

00:00:47.790 --> 00:00:51.180
So this is the change in
f over the change in x.

00:00:51.180 --> 00:00:53.460
So we see over
here, when x changes

00:00:53.460 --> 00:00:59.930
by 1, the value of our
function changes by positive 5.

00:01:02.650 --> 00:01:05.310
And it's linear, so that's true.

00:01:05.310 --> 00:01:07.360
Between any two
points, the ratio

00:01:07.360 --> 00:01:10.410
between our change in f and
our change in x is the same.

00:01:10.410 --> 00:01:15.710
If we go up 1 again, we have
plus 1 in the x-direction,

00:01:15.710 --> 00:01:18.302
we are once again
increasing by 5.

00:01:18.302 --> 00:01:20.510
If you start from this point
and go all the way here,

00:01:20.510 --> 00:01:24.960
so if you go plus
2 along x, you're

00:01:24.960 --> 00:01:29.040
going to go plus 10 along f.

00:01:29.040 --> 00:01:31.590
So it would be 10 over
2, which is still 5.

00:01:31.590 --> 00:01:33.930
So either way, the
slope, or the rate

00:01:33.930 --> 00:01:36.090
of change of the vertical
axis with respect

00:01:36.090 --> 00:01:39.630
to the horizontal
axis, is 5 for f.

00:01:39.630 --> 00:01:42.790
Now, let's see which of
these increase faster.

00:01:42.790 --> 00:01:45.110
Well, a isn't even increasing.

00:01:45.110 --> 00:01:46.730
So A is decreasing.

00:01:46.730 --> 00:01:49.620
As x increases, y is decreasing.

00:01:49.620 --> 00:01:53.270
So that definitely
can't be the case.

00:01:53.270 --> 00:01:55.610
If we look at this
one right over here,

00:01:55.610 --> 00:01:59.790
it looks like-- let's see,
if we start over here,

00:01:59.790 --> 00:02:04.540
if we increase 1 along the
x-direction, if our change in x

00:02:04.540 --> 00:02:08.120
is 1, it looks like our
change in y is exactly 5-- 1,

00:02:08.120 --> 00:02:11.210
2, 3, 4, 5.

00:02:11.210 --> 00:02:16.410
So it looks like for choice
B, our slope is exactly

00:02:16.410 --> 00:02:19.940
5, or our change in y over
change in x is exactly 5.

00:02:19.940 --> 00:02:21.820
So it's not increasing
faster than f.

00:02:21.820 --> 00:02:24.060
It's increasing the same as f.

00:02:24.060 --> 00:02:26.040
Now let's look at
C. So I'm going

00:02:26.040 --> 00:02:29.494
to try to find a point where
it looks like I have an integer

00:02:29.494 --> 00:02:30.410
point right over here.

00:02:30.410 --> 00:02:33.610
So that's the point,
negative 3, negative 3.

00:02:33.610 --> 00:02:36.580
And if I move 1 in
the x-direction,

00:02:36.580 --> 00:02:38.980
it looks like I'm
increasing by more than 5.

00:02:38.980 --> 00:02:47.650
I'm increasing 1, 2, 3, 4,
5, 6, 7, 8, it looks like.

00:02:47.650 --> 00:02:51.320
So this one looks like
it has a slope of 8.

00:02:51.320 --> 00:02:55.110
So this one is
increasing faster than f,

00:02:55.110 --> 00:02:56.924
so we'll circle that
right over there.

00:02:56.924 --> 00:02:58.340
And now let's look
at this choice.

00:02:58.340 --> 00:03:00.370
So if we start right over here--
and I just picked this point

00:03:00.370 --> 00:03:02.500
because that's at a
nice integer coordinate.

00:03:02.500 --> 00:03:04.625
It's at the point 2, negative 4.

00:03:09.320 --> 00:03:13.310
If we increase x by 1,
then we increase y by 1,

00:03:13.310 --> 00:03:17.850
2, 3-- looks like about 3
and 1/2, definitely not 5.

00:03:17.850 --> 00:03:19.790
In order for it to
increase as fast as f,

00:03:19.790 --> 00:03:23.000
it would have to increase by 5,
so it would have to be up here.

00:03:23.000 --> 00:03:24.995
So it would have to
go 1, 2, 3, 4, 5.

00:03:24.995 --> 00:03:26.620
It would have had to
have been up here.

00:03:26.620 --> 00:03:30.990
The line would've looked
something more like that just

00:03:30.990 --> 00:03:34.030
to match f, much less
grow faster than f.

00:03:34.030 --> 00:03:36.230
So D does not meet the criteria.

00:03:36.230 --> 00:03:38.463
It is only C.


Comparing linear functions word problems 2

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.990
Elizabeth starts out 5
miles away from school

00:00:04.990 --> 00:00:08.880
and walks away from school
at 3 miles per hour.

00:00:08.880 --> 00:00:10.330
So she's already 5 miles away.

00:00:10.330 --> 00:00:12.000
And she's going to
walk even further

00:00:12.000 --> 00:00:14.570
away at 3 miles per hour.

00:00:14.570 --> 00:00:17.680
The table below shows how
far some other students

00:00:17.680 --> 00:00:20.660
are from school
at various times.

00:00:20.660 --> 00:00:25.270
Each person is moving at a
constant speed starting at time

00:00:25.270 --> 00:00:27.780
is equal to 0.

00:00:27.780 --> 00:00:31.020
Which students started
out farther from school

00:00:31.020 --> 00:00:31.930
than Elizabeth?

00:00:31.930 --> 00:00:33.939
Select all that apply.

00:00:33.939 --> 00:00:35.480
So essentially, we
need to figure out

00:00:35.480 --> 00:00:38.060
where these students
were at time equals 0.

00:00:38.060 --> 00:00:41.940
So we know where they
were at time 1, 2, and 3.

00:00:41.940 --> 00:00:43.840
And so let's think
about their rate

00:00:43.840 --> 00:00:46.660
towards or away from school.

00:00:46.660 --> 00:00:50.590
And remember, this is
distance from school.

00:00:50.590 --> 00:00:54.680
As we increase-- as we
go from hour 1 to hour 2,

00:00:54.680 --> 00:00:57.420
Gordon gets 2 miles closer.

00:00:57.420 --> 00:01:01.410
So his distance to
school is decreasing.

00:01:01.410 --> 00:01:03.916
So where was he
at time equals 0?

00:01:03.916 --> 00:01:06.290
I'll put time equals 0 up here
because I don't have any--

00:01:06.290 --> 00:01:07.270
actually, I'll
put it right here.

00:01:07.270 --> 00:01:09.080
I'll try to squeeze
it into the chart.

00:01:09.080 --> 00:01:11.990
So where was he
at time equals 0?

00:01:11.990 --> 00:01:15.640
Well, he would have
been 2 miles further.

00:01:15.640 --> 00:01:18.370
So he would have
been 6 miles away.

00:01:18.370 --> 00:01:19.870
Notice that it's consistent.

00:01:19.870 --> 00:01:22.270
In the first hour,
he would have gotten

00:01:22.270 --> 00:01:24.145
2 miles closer to school.

00:01:24.145 --> 00:01:25.520
Then the next
hour, he would have

00:01:25.520 --> 00:01:27.200
gotten 2 miles even closer.

00:01:27.200 --> 00:01:29.720
And then the third hour, he
actually gets 2 miles closer.

00:01:29.720 --> 00:01:31.490
And he actually gets to school.

00:01:31.490 --> 00:01:36.860
So Gordon started out 6
miles away at t equals 0.

00:01:36.860 --> 00:01:39.970
So Gordon did start out farther
from school than Elizabeth.

00:01:39.970 --> 00:01:41.620
So we can circle Gordon.

00:01:41.620 --> 00:01:43.200
He meets the conditions.

00:01:43.200 --> 00:01:45.730
Now let's think about Giovanni.

00:01:45.730 --> 00:01:49.840
So at time 1, he's 5
miles away from school.

00:01:49.840 --> 00:01:54.150
Then at 1 hour,
he's 5 miles away.

00:01:54.150 --> 00:01:56.890
After 2 hours,
he's 6 miles away.

00:01:56.890 --> 00:01:59.350
So he's getting
further from school.

00:01:59.350 --> 00:02:01.410
So this is a plus 1.

00:02:01.410 --> 00:02:04.130
And then after another
hour, he is 7 miles away.

00:02:04.130 --> 00:02:07.790
So every hour that goes
by, he's a mile further.

00:02:07.790 --> 00:02:11.110
He's going 1 mile an
hour away from school.

00:02:11.110 --> 00:02:14.640
So where was he at time equal 0?

00:02:14.640 --> 00:02:17.110
Well, he would have been
a mile closer to school

00:02:17.110 --> 00:02:18.750
relative to time equal 1.

00:02:18.750 --> 00:02:21.470
So he would have
been 4 miles away.

00:02:21.470 --> 00:02:23.880
So he did not start out
farther than Elizabeth,

00:02:23.880 --> 00:02:26.540
who started out 5 miles away.

00:02:26.540 --> 00:02:28.350
Now let's look at Hannah.

00:02:28.350 --> 00:02:32.880
Hannah, at every time, is
just exactly 5 miles away

00:02:32.880 --> 00:02:34.070
from school.

00:02:34.070 --> 00:02:35.840
So she's napping or something.

00:02:35.840 --> 00:02:39.050
She is not actually moving.

00:02:39.050 --> 00:02:43.570
She started out napping at
exactly the same distance

00:02:43.570 --> 00:02:47.075
as Elizabeth, but she did not
start out farther from school

00:02:47.075 --> 00:02:47.700
than Elizabeth.

00:02:47.700 --> 00:02:51.470
So Hannah does not
meet the criteria.

00:02:51.470 --> 00:02:53.050
Now let's look at Alberto.

00:02:53.050 --> 00:02:58.370
At time equals 1, he
is 9 miles from school.

00:02:58.370 --> 00:03:04.580
And then after 1 hour, he
gets a mile and 1/2 further

00:03:04.580 --> 00:03:05.870
from school.

00:03:05.870 --> 00:03:10.540
After another hour, he gets
a mile and 1/2 even further.

00:03:10.540 --> 00:03:12.796
So where was he
at time equals 0?

00:03:12.796 --> 00:03:15.170
Well, he would have been a
mile and 1/2 closer to school.

00:03:15.170 --> 00:03:20.680
So 9 minus 1.5 is-- he would
have been 7 and 1/2 miles away.

00:03:20.680 --> 00:03:24.200
So even though he is
going away from-- well,

00:03:24.200 --> 00:03:28.810
he definitely started further
from school than Elizabeth.

00:03:28.810 --> 00:03:30.540
Elizabeth started
out 5 miles away.

00:03:30.540 --> 00:03:33.200
Alberto started off
7 and 1/2 miles away

00:03:33.200 --> 00:03:35.820
and is going even further
and further and further.

00:03:35.820 --> 00:03:38.070
So the two students that
start out farther from school

00:03:38.070 --> 00:03:42.350
than Elizabeth are
Gordon and Alberto.


Comparing linear functions 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.420
Two functions, f and
g, are described below.

00:00:04.420 --> 00:00:08.260
Which of these statements
about f and g is true?

00:00:08.260 --> 00:00:10.720
So they defined
function f as kind

00:00:10.720 --> 00:00:13.560
of a traditional linear
equation right over here.

00:00:13.560 --> 00:00:16.070
And this right over here is g.

00:00:16.070 --> 00:00:19.150
So this right over
here is g of x.

00:00:19.150 --> 00:00:21.530
And that also looks
like a linear function.

00:00:21.530 --> 00:00:24.080
We see it's a kind of a
downward sloping line.

00:00:24.080 --> 00:00:27.390
So let's look at our choices
and see which of these are true.

00:00:27.390 --> 00:00:32.299
f and g are both increasing, and
f is increasing faster than g.

00:00:32.299 --> 00:00:34.960
Well, when I look at g--
Well, first of all, g

00:00:34.960 --> 00:00:36.390
is definitely decreasing.

00:00:36.390 --> 00:00:37.970
So we already know
that that's false.

00:00:37.970 --> 00:00:39.660
And f is also decreasing.

00:00:39.660 --> 00:00:41.920
We see here it has
a negative slope.

00:00:41.920 --> 00:00:45.620
Every time we move forward
3 in the x direction,

00:00:45.620 --> 00:00:49.180
we're going to move down 7
in the vertical direction.

00:00:49.180 --> 00:00:51.730
So neither of these
are increasing

00:00:51.730 --> 00:00:54.080
so that's definitely not right.

00:00:54.080 --> 00:00:55.400
f and g are both increasing.

00:00:55.400 --> 00:00:57.640
Well, that's
definitely not right.

00:00:57.640 --> 00:01:00.550
So we know that both f
and g are decreasing.

00:01:00.550 --> 00:01:03.390
So this first choice says
they're both decreasing, and g

00:01:03.390 --> 00:01:06.400
is decreasing faster than f.

00:01:06.400 --> 00:01:09.160
So let's see what
the slope on g is.

00:01:09.160 --> 00:01:14.380
So the slope on g is every time
we move 1 in the x direction,

00:01:14.380 --> 00:01:16.510
positive 1 in the
x direction, we

00:01:16.510 --> 00:01:21.460
move down 2 in the y direction.

00:01:21.460 --> 00:01:25.990
So for g of x, if we were
to write our change in y

00:01:25.990 --> 00:01:29.750
over our change in x-- which
is our slope-- our change in y

00:01:29.750 --> 00:01:33.030
over change in x, when we
move one in the x direction,

00:01:33.030 --> 00:01:35.060
positive 1 in the
x direction, we

00:01:35.060 --> 00:01:38.120
move down 2 in the y direction.

00:01:38.120 --> 00:01:42.960
So our change in y over
change in x is negative 2.

00:01:42.960 --> 00:01:46.420
So g has a slope of negative 2.

00:01:46.420 --> 00:01:49.300
f has a slope of negative 7/3.

00:01:49.300 --> 00:01:56.120
Negative 7/3 is the same
thing as negative 2 and 1/3.

00:01:56.120 --> 00:01:58.870
So f's slope is more negative.

00:01:58.870 --> 00:02:01.070
So it is decreasing faster.

00:02:01.070 --> 00:02:04.520
So g is not decreasing faster
than f. f is decreasing faster

00:02:04.520 --> 00:02:05.020
than g.

00:02:05.020 --> 00:02:06.750
So this is not right.

00:02:06.750 --> 00:02:09.440
And then we have this choice--
f and g are both decreasing,

00:02:09.440 --> 00:02:12.010
and f is decreasing
faster than g.

00:02:12.010 --> 00:02:13.960
This is right, right over here.

00:02:13.960 --> 00:02:17.870
We have this last choice-- g is
increasing but f is decreasing.

00:02:17.870 --> 00:02:21.670
We know that's not true.
g is actually decreasing.


Systems of nonlinear equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.280
Which of the following graphs
represents the solutions

00:00:04.280 --> 00:00:10.170
to the system of equations x
is equal to 2 minus y and y

00:00:10.170 --> 00:00:13.010
is equal to x squared minus 4.

00:00:13.010 --> 00:00:15.170
So essentially, we
want to find the graphs

00:00:15.170 --> 00:00:17.500
that represent
these two equations,

00:00:17.500 --> 00:00:20.710
and where they intersect
represent the solutions.

00:00:20.710 --> 00:00:23.180
So, let's see. y equals
x squared minus 4.

00:00:23.180 --> 00:00:24.690
That's going to be
an upward opening

00:00:24.690 --> 00:00:28.000
parabola with a
y-intercept of negative 4,

00:00:28.000 --> 00:00:29.340
and we see that is the case.

00:00:29.340 --> 00:00:32.189
Or I should say the
y-coordinate of its vertex

00:00:32.189 --> 00:00:34.950
would be negative 4 right
over there, and also

00:00:34.950 --> 00:00:36.080
the y-intercept.

00:00:36.080 --> 00:00:38.304
So that looks like this
one right over here.

00:00:38.304 --> 00:00:40.220
And this is the only one
that opens like that,

00:00:40.220 --> 00:00:42.178
so we can immediately
say that this is probably

00:00:42.178 --> 00:00:44.710
going to be the red
one right over here.

00:00:44.710 --> 00:00:51.260
But we can verify that this line
right over here is 2 minus x.

00:00:51.260 --> 00:00:56.090
So if I were to write this in
traditional slope intercept

00:00:56.090 --> 00:01:01.500
form, I could rewrite this
as x is equal to 2 minus y.

00:01:01.500 --> 00:01:02.680
Add y to both sides.

00:01:02.680 --> 00:01:06.040
We have y plus x
is equal to 2, or y

00:01:06.040 --> 00:01:08.380
is equal to negative x plus 2.

00:01:08.380 --> 00:01:13.190
So we see the y-intercept is
2, and its slope is negative 1.

00:01:13.190 --> 00:01:16.410
So that's indeed this line,
and so the solutions here

00:01:16.410 --> 00:01:19.470
are going to be that point and
that point right over here.

00:01:19.470 --> 00:01:23.230
Red is what represents
the solutions

00:01:23.230 --> 00:01:26.945
to this non-linear
system of equations.


Systems of nonlinear equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.460
What are the solutions to
the system of equations

00:00:03.460 --> 00:00:07.160
y is equal to x plus
1, and x squared

00:00:07.160 --> 00:00:09.680
plus y squared is equal to 25?

00:00:09.680 --> 00:00:12.040
So let's first just visualize
what we're trying to do.

00:00:12.040 --> 00:00:17.430
So let me try to roughly
graph these two equations.

00:00:17.430 --> 00:00:20.293
So my y-axis, this is my x-axis.

00:00:23.480 --> 00:00:25.730
This right over here, x
squared plus y squared

00:00:25.730 --> 00:00:28.650
is equal to 25, that's going
to be a circle centered

00:00:28.650 --> 00:00:31.667
at 0 with radius 5.

00:00:31.667 --> 00:00:33.750
You don't have to know
that to solve this problem,

00:00:33.750 --> 00:00:35.080
but it helps to visualize it.

00:00:35.080 --> 00:00:39.130
So if this is 5,
this is 5, 5, 5.

00:00:39.130 --> 00:00:41.360
This right over
here is negative 5.

00:00:41.360 --> 00:00:43.720
This right over
here is negative 5.

00:00:43.720 --> 00:00:45.480
This equation would
be represented

00:00:45.480 --> 00:00:46.900
by this set of
points, or this is

00:00:46.900 --> 00:00:50.160
a set of points that
satisfy this equation.

00:00:50.160 --> 00:00:52.330
So let me-- there you go.

00:00:52.330 --> 00:00:55.810
Trying to draw it as close
to a perfect circle as I can.

00:00:55.810 --> 00:01:01.700
And then y equals
x plus 1 is a line

00:01:01.700 --> 00:01:04.670
of slope 1 with a 1 y-intercept.

00:01:04.670 --> 00:01:07.240
So this is 1, 2, 3, 4.

00:01:07.240 --> 00:01:09.850
y-intercept is there
and has a slope of 1

00:01:09.850 --> 00:01:13.224
so it looks something like this.

00:01:13.224 --> 00:01:14.890
So when we're looking
for the solutions,

00:01:14.890 --> 00:01:17.020
we're looking for the
points that satisfy both.

00:01:17.020 --> 00:01:19.630
The points that satisfy both
are the points that sit on both.

00:01:19.630 --> 00:01:22.870
So it's that point-- let me do
it in green-- It's this point

00:01:22.870 --> 00:01:25.680
and it's this point
right over here.

00:01:25.680 --> 00:01:28.100
So how do we actually
figure that out?

00:01:28.100 --> 00:01:30.320
Well, the easiest way
is to-- well, sometimes

00:01:30.320 --> 00:01:32.960
the easiest way is to substitute
one of these constraints

00:01:32.960 --> 00:01:34.710
into the other constraint.

00:01:34.710 --> 00:01:36.840
And since they've already
solved for y here,

00:01:36.840 --> 00:01:40.740
we can substitute y in the
blue equation with x plus 1

00:01:40.740 --> 00:01:42.580
with this constraint
right over here.

00:01:42.580 --> 00:01:45.690
So instead of saying x squared
plus y squared equals 25,

00:01:45.690 --> 00:01:49.880
we can say x squared plus,
and instead of writing a y,

00:01:49.880 --> 00:01:53.310
we're adding the constraint
that y must be x plus 1.

00:01:53.310 --> 00:02:02.960
So x squared plus x plus 1
squared must be equal to 25.

00:02:02.960 --> 00:02:06.010
And now, we can
attempt to solve for x.

00:02:06.010 --> 00:02:09.539
So we get x squared
plus-- now we square this.

00:02:09.539 --> 00:02:11.340
We'll get-- let me
write in magenta--

00:02:11.340 --> 00:02:15.060
we'll get x squared
plus 2x plus 1,

00:02:15.060 --> 00:02:18.380
and that must be equal to 25.

00:02:18.380 --> 00:02:20.890
We have 2x squared--
now, I'm just

00:02:20.890 --> 00:02:23.340
combining these two
terms-- 2x squared

00:02:23.340 --> 00:02:28.190
plus 2x plus 1 is equal to 25.

00:02:28.190 --> 00:02:30.250
Now, we could just use
the quadratic formula

00:02:30.250 --> 00:02:33.760
to find the-- well,
we have to be careful.

00:02:33.760 --> 00:02:35.540
We have to set this
equal to 0, and then

00:02:35.540 --> 00:02:37.180
use the quadratic formula.

00:02:37.180 --> 00:02:39.430
So let's subtract
25 from both sides,

00:02:39.430 --> 00:02:46.406
and you could get 2x squared
plus 2x minus 24 is equal to 0.

00:02:46.406 --> 00:02:48.280
And actually, let's--
just to simplify this--

00:02:48.280 --> 00:02:53.710
let's divide both sides by 2,
and you get x squared plus x

00:02:53.710 --> 00:02:56.027
minus 12 is equal to 0.

00:02:56.027 --> 00:02:58.610
And actually, we don't even have
to use the quadratic formula,

00:02:58.610 --> 00:03:00.474
we can factor this
right over here.

00:03:00.474 --> 00:03:02.640
What are two numbers that
when we take their product

00:03:02.640 --> 00:03:06.880
we get negative 12, and when
we add them, we get positive 1?

00:03:06.880 --> 00:03:09.820
Well, positive 4 and negative
3 would do the trick.

00:03:09.820 --> 00:03:16.950
So we have x plus 4 times
x minus 3 is equal to 0.

00:03:16.950 --> 00:03:21.220
So x could be equal
to-- well, if x plus 4

00:03:21.220 --> 00:03:23.340
is 0 then that would make
this whole thing true.

00:03:23.340 --> 00:03:26.930
So x could be equal
to negative 4 or x

00:03:26.930 --> 00:03:30.180
could be equal to positive 3.

00:03:30.180 --> 00:03:31.990
So this right over
here is a situation

00:03:31.990 --> 00:03:33.740
where x is negative 4.

00:03:33.740 --> 00:03:36.690
This right over here is
a situation where x is 3.

00:03:36.690 --> 00:03:39.490
So we're almost done, we just
have to find the corresponding

00:03:39.490 --> 00:03:40.340
y's.

00:03:40.340 --> 00:03:43.270
And for that, we can just
resort to the simplest equation

00:03:43.270 --> 00:03:45.470
right over here, y is x plus 1.

00:03:45.470 --> 00:03:48.360
So in this situation
when x is negative 4,

00:03:48.360 --> 00:03:50.770
y is going to be that plus 1.

00:03:50.770 --> 00:03:55.370
So y is going to be negative 3.

00:03:55.370 --> 00:03:59.660
This is the point negative
4 comma negative 3.

00:03:59.660 --> 00:04:03.970
Likewise, when x is 3, y
is going to be equal to 4.

00:04:03.970 --> 00:04:07.150
So this is the point 3 comma 4.

00:04:07.150 --> 00:04:11.440
These are the two solutions
to this non-linear system

00:04:11.440 --> 00:04:12.991
of equations.


Systems of nonlinear equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:06.010
What are the solutions to
the system of equations y

00:00:06.010 --> 00:00:13.610
is equal to 1/2 x and 2x squared
minus y squared is equal to 7.

00:00:13.610 --> 00:00:16.460
And they say solutions,
because these two,

00:00:16.460 --> 00:00:19.750
if we view them as two
curves, they could very well

00:00:19.750 --> 00:00:22.154
intersect with each other twice.

00:00:22.154 --> 00:00:23.570
So let's see what's
going on here.

00:00:23.570 --> 00:00:29.530
We have y is equal to 1/2 x
and 2x squared minus y squared

00:00:29.530 --> 00:00:30.804
is equal to 7.

00:00:30.804 --> 00:00:32.220
And the best way
to approach these

00:00:32.220 --> 00:00:34.620
is to just try to
substitute one constraint

00:00:34.620 --> 00:00:36.410
into the other
constraint, or substitute

00:00:36.410 --> 00:00:38.680
one equation into the other one.

00:00:38.680 --> 00:00:44.340
It seems easier to substitute
1/2 x for y into this equation,

00:00:44.340 --> 00:00:47.100
because they've already
solved for y here.

00:00:47.100 --> 00:00:50.590
Here, it's much harder to solve
for x or y, so let's do that.

00:00:50.590 --> 00:00:54.520
Every place we see a y here,
let's substitute it with this,

00:00:54.520 --> 00:00:57.650
that y must also
be equal to 1/2 x,

00:00:57.650 --> 00:01:00.470
and then see if we
can solve for x.

00:01:00.470 --> 00:01:06.290
So on this equation, we have
2x squared minus y squared.

00:01:06.290 --> 00:01:13.240
But now we're saying that y
must also be equal to 1/2 x.

00:01:13.240 --> 00:01:16.720
And that is going
to be equal to 7.

00:01:16.720 --> 00:01:18.380
Now let's see if
we can solve for x

00:01:18.380 --> 00:01:21.080
doing a little bit of
algebraic manipulation.

00:01:21.080 --> 00:01:24.230
So we get 2x squared minus.

00:01:24.230 --> 00:01:27.500
So 1/2 squared is 1/4,
and then x squared.

00:01:27.500 --> 00:01:30.760
So we could say x squared
over-- let me write that as 1/4.

00:01:30.760 --> 00:01:35.980
So let's say it's 1/4 x
squared is equal to 7.

00:01:35.980 --> 00:01:39.710
So I have 2x squareds, and I
subtract out a 1/4 x squared,

00:01:39.710 --> 00:01:42.580
so I'm going to have
a 1 and 3/4 x squared.

00:01:42.580 --> 00:01:47.610
Or you could view this as 8/4
minus 1/4 is 7/4 x squared.

00:01:47.610 --> 00:01:50.980
7/4 x squared is equal to 7.

00:01:50.980 --> 00:01:55.594
Multiply both sides times the
reciprocal of 7/4, so 4/7.

00:01:58.320 --> 00:02:00.890
Multiply both sides by 4/7.

00:02:00.890 --> 00:02:05.810
And we get x squared
is equal to 4.

00:02:09.250 --> 00:02:13.140
And so x could be
positive or negative 2.

00:02:13.140 --> 00:02:15.550
It's the positive and
negative square root of 4.

00:02:15.550 --> 00:02:20.140
So x is equal to the plus
or minus square root of 4.

00:02:20.140 --> 00:02:26.913
x is equal to positive
2 or negative 2.

00:02:26.913 --> 00:02:31.020
Now, given that x is
positive 2 or negative 2,

00:02:31.020 --> 00:02:34.070
let's substitute back into
either of these equations

00:02:34.070 --> 00:02:36.540
to figure out what y is,
what the corresponding y

00:02:36.540 --> 00:02:38.040
is for each of these.

00:02:38.040 --> 00:02:41.190
So if x is 2, y is
going to be 1/2 of that.

00:02:41.190 --> 00:02:42.570
It's going to be 1.

00:02:42.570 --> 00:02:45.270
So we have the point 2 comma 1.

00:02:45.270 --> 00:02:47.120
All I did is say, look, x is 2.

00:02:47.120 --> 00:02:48.990
1/2 times 2 is 1.

00:02:48.990 --> 00:02:53.610
If x is negative
2, then y is going

00:02:53.610 --> 00:02:56.000
to be 1/2 times
negative 2, which

00:02:56.000 --> 00:02:58.020
is going to be negative 1.

00:02:58.020 --> 00:03:01.430
And both of these definitely
satisfy this first constraint.

00:03:01.430 --> 00:03:03.140
And you can verify
that they also

00:03:03.140 --> 00:03:05.440
satisfy this second
constraint right over here.

00:03:05.440 --> 00:03:09.410
2 times 2 squared is--
well, 2 squared is 4.

00:03:09.410 --> 00:03:11.250
2 times that is 8.

00:03:11.250 --> 00:03:14.130
Minus 1 squared is 7.

00:03:14.130 --> 00:03:15.875
2 times negative 2
squared is still--

00:03:15.875 --> 00:03:17.250
this whole thing's
going to be 8.

00:03:17.250 --> 00:03:20.060
Minus negative 1
squared still equals 7.

00:03:20.060 --> 00:03:22.880
So the solutions to the
system of equation-- one

00:03:22.880 --> 00:03:26.176
is the coordinate 2
comma 1. x is 2. y is 1.

00:03:26.176 --> 00:03:27.800
The other is the
coordinate negative 2,

00:03:27.800 --> 00:03:33.310
negative 1. x is negative
2, and y is negative 1.


Comparing linear functions 2

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:04.847
f is a linear function whose
table of values is shown below.

00:00:04.847 --> 00:00:07.430
So they give us different values
of x and what the function is

00:00:07.430 --> 00:00:09.250
for each of those x's.

00:00:09.250 --> 00:00:11.210
Which graphs show
functions which

00:00:11.210 --> 00:00:14.860
are increasing at
the same rate as f?

00:00:14.860 --> 00:00:17.900
So what is the rate at
which f is increasing?

00:00:17.900 --> 00:00:24.700
When x increases by 4, we have
our function increasing by 7.

00:00:24.700 --> 00:00:27.140
So we could just look
for which of these lines

00:00:27.140 --> 00:00:32.500
are increasing at a rate of
7/4, 7 in the vertical direction

00:00:32.500 --> 00:00:35.780
every time we move 4 in
the horizontal direction.

00:00:35.780 --> 00:00:38.820
And an easy way to eyeball
that would actually be just

00:00:38.820 --> 00:00:43.670
to plot two points for f, and
then see what that rate looks

00:00:43.670 --> 00:00:45.070
like visually.

00:00:45.070 --> 00:00:48.390
So if we see here when
x is 0, f is negative 1.

00:00:48.390 --> 00:00:53.760
When x is 0, f is negative 1.

00:00:53.760 --> 00:00:57.210
So when x is 0, f is negative 1.

00:00:57.210 --> 00:01:08.010
And when x is 4, f is
6, so 1, 2, 3, 4, 5, 6,

00:01:08.010 --> 00:01:09.480
so just like that.

00:01:09.480 --> 00:01:11.350
And two points specify a line.

00:01:11.350 --> 00:01:13.924
We know that it is
a linear function.

00:01:13.924 --> 00:01:15.090
You can even verify it here.

00:01:15.090 --> 00:01:18.660
When we increase by 4 again,
we increase our function by 7

00:01:18.660 --> 00:01:19.280
again.

00:01:19.280 --> 00:01:20.655
We know that these
two points are

00:01:20.655 --> 00:01:24.790
on f and so we get a sense
of the rate of change of f.

00:01:24.790 --> 00:01:27.290
Now, when you draw it
like that, it immediately

00:01:27.290 --> 00:01:30.160
becomes pretty
clear which of these

00:01:30.160 --> 00:01:33.290
has the same rate
of change of f.

00:01:33.290 --> 00:01:35.510
A is increasing faster than f.

00:01:35.510 --> 00:01:39.980
C is increasing slower.

00:01:39.980 --> 00:01:41.690
A is increasing
much faster than f.

00:01:41.690 --> 00:01:43.800
C is increasing slower than f.

00:01:43.800 --> 00:01:46.580
B is decreasing, so
that's not even close.

00:01:46.580 --> 00:01:49.970
But D seems to have the
exact same inclination,

00:01:49.970 --> 00:01:53.970
the exact same slope, as f.

00:01:53.970 --> 00:01:56.100
So D is what we would go with.

00:01:56.100 --> 00:01:57.480
And we could even
verify it, even

00:01:57.480 --> 00:02:00.360
if we didn't draw
it in this way.

00:02:00.360 --> 00:02:05.830
Our change in f for
a given change in x

00:02:05.830 --> 00:02:09.590
is equal to-- when
x changed plus 4,

00:02:09.590 --> 00:02:11.700
our function changed plus 7.

00:02:11.700 --> 00:02:14.090
It is equal to 7/4.

00:02:14.090 --> 00:02:18.690
And we can verify that on D, if
we increase in the x-direction

00:02:18.690 --> 00:02:24.220
by 4, so we go from 4 to 8,
then in the vertical direction

00:02:24.220 --> 00:02:30.160
we should increase by 7,
so 1, 2, 3, 4, 5, 6, 7.

00:02:30.160 --> 00:02:34.794
And it, indeed, does increase
at the exact same rate.


How to find the average rate of change from a graph (example)

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:05.050
Over which interval does y
of x have an average rate

00:00:05.050 --> 00:00:07.779
of change of negative 4?

00:00:07.779 --> 00:00:09.820
So average rate of change,
if you think about it,

00:00:09.820 --> 00:00:11.250
you are literally just
averaging for example,

00:00:11.250 --> 00:00:13.110
in this bowl section
right over here.

00:00:13.110 --> 00:00:14.700
The slope is really,
really steep.

00:00:14.700 --> 00:00:15.690
It gets less steep.

00:00:15.690 --> 00:00:18.580
It's a very negative slope,
it gets less negative.

00:00:18.580 --> 00:00:21.290
Less negative slope is 0 here.

00:00:21.290 --> 00:00:23.830
Then it gets positive, more
positive, and more positive,

00:00:23.830 --> 00:00:25.080
and more positive.

00:00:25.080 --> 00:00:27.038
But when you get to this
point right over here,

00:00:27.038 --> 00:00:30.530
you see you got to
where you started from.

00:00:30.530 --> 00:00:34.950
One you could say the
net change has been 0.

00:00:34.950 --> 00:00:39.060
And any interval over which
the net change has been 0 also

00:00:39.060 --> 00:00:42.470
tells you that the average rate
of change is going to be 0.

00:00:42.470 --> 00:00:45.640
So you could view that
the average rate of change

00:00:45.640 --> 00:00:49.550
is really the slope
of the line that

00:00:49.550 --> 00:00:53.250
connects the two endpoints
of your interval.

00:00:53.250 --> 00:00:55.570
So another way of asking
over which interval

00:00:55.570 --> 00:00:58.430
does y of x have an average
rate of change of negative 4

00:00:58.430 --> 00:01:00.350
is, can you come
up with an interval

00:01:00.350 --> 00:01:03.330
where the slope between the
endpoints of the interval

00:01:03.330 --> 00:01:05.447
is negative 4?

00:01:05.447 --> 00:01:07.030
So let's see the
choices they give us.

00:01:07.030 --> 00:01:11.040
This first interval is x is
between negative 1 and 1.

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So x is between negative 1.

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So this is x is negative 1.

00:01:14.920 --> 00:01:18.700
When x is equal to negative 1,
y of x is all the way over here.

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y of negative 1 is equal to 7.

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And then when x is equal to 1,
our graph is down over here.

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y of 1 is negative 1.

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So what is the slope
of the line that

00:01:33.830 --> 00:01:38.090
connects the endpoints
of those two points?

00:01:38.090 --> 00:01:39.910
So what is the
slope of this line?

00:01:39.910 --> 00:01:41.850
Because the slope of
this line, the line that

00:01:41.850 --> 00:01:44.000
connects the endpoints
of my interval,

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that is going to be the
average rate of change

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over this interval.

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And you see very clearly
that the slope here,

00:01:50.540 --> 00:01:54.960
the rate of change of y with
respect to x is negative 4.

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Every time we move one
ahead in the x direction,

00:01:57.660 --> 00:02:00.200
we move down four
in the y direction.

00:02:00.200 --> 00:02:02.050
Move one ahead in
the x direction,

00:02:02.050 --> 00:02:05.580
we move down four
in the y direction.

00:02:05.580 --> 00:02:08.800
So the average rate of
change over this interval

00:02:08.800 --> 00:02:09.949
is negative 4.

00:02:09.949 --> 00:02:12.000
So we didn't have to even
look at anything else,

00:02:12.000 --> 00:02:14.380
that one will work.