Finding 1 on the number line

WEBVTT
Kind: captions
Language: en

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- [Instructor] I'm here at
the Khan Academy exercise

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called Find One on the Number Line

00:00:05.300 --> 00:00:07.190
and they're asking us to do exactly that.

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It says, "Move the dot to
one on the number line,"

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and it's a little interactive dot

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that I can move around.

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And so, let's think
about how I would do it

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and I always encourage you,

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pause this video and see at least how

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you would think about doing it

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or put your finger on the
screen where you think one is

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and then we'll work through it together.

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All right.

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So they've told us where zero is

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and they've told us
where seven fourths is.

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So one thing I could do is

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I could say well how many
of these equal spaces

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does it take me to get
from zero to seven fourths.

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So let's see it's one, two,
three, four, five, six, seven.

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So if it takes seven equal
spaces to get to seven fourths,

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that means that each of
these spaces is one fourth

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because then it would be
one fourth, two fourths,

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three fourths, four fourths,
five fourths, six fourths,

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and seven fourths.

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All right, we're making
some good progress.

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So where would one be?

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Well, one would be four fourths.

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So we would go one fourth,
two fourths, three fourths,

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and then four fourths.

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So that's where one is.

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So the important thing to
realize is the only way

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we knew that each of
these, each of these gaps,

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or each of these spaces
from one hash to the next,

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the way that we knew that
each of those is a fourth

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is by saying hey look,
seven of those equal spaces

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get us to seven fourths,

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so each of these must be a fourth.

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So four of those, four
fourths, would be equal to one.

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Let's do another example.

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So here, we're said,
we're told to move the dot

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to one on the number line.

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So put your finger on where
that would be on the screen.

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Pause the video and do that.

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All right.

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In some ways this is a little bit easier

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because they told us that going from zero

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to the next little cross or
hash, I guess you could say,

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whatever you want to call
these things, is one sixth.

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So what is one?

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Well, one or one whole is six sixths.

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So this is one sixth,
two sixths, three sixths,

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four sixths, five sixths,
and then six sixths.

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So there we go.

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That is where is one
is on this number line.


Relating number lines to fraction bars

WEBVTT
Kind: captions
Language: en

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- [Instructor] We are asked
what fraction is located

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at point A on the number line?

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And we can see point A right there.

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Pause this video and see
if could answer that.

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All right, now there's a bunch of ways

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that you could think about it.

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You could see that the
space between zero and one

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is split into one, two,
three, four equal spaces.

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And this has gone three
of those four equal spaces

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from zero to one.

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So that's one interesting
way to think about it.

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Another thing that might help us

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is a bit of a visualization.

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If this rectangle represents a whole

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and notice it goes from zero to one,

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so you could view one as a whole,

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we have split it into four equal sections.

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So each of these equal sections
you would consider a fourth.

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So that's a fourth right over there.

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That's another fourth right over there.

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This is another fourth right over there.

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So how many of these
fourths have been shaded in?

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Well three of them have been shaded in.

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And when you look at the number line,

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you see the same idea.

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When we see the space between zero and one

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it has been split into fourths.

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So this is a fourth,
and then another fourth,

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and then another fourth,
and another fourth.

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And where is point A?

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Well we have gone 1/4, 2/4, 3/4 past zero

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or from zero to one, which is a whole.

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So what fraction is located
at point A on the number line?

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3/4.

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Let's do another example.

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So here we're told which point
is at 2/6 on the number line.

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Pause this video and see if
you can answer on your own

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before we work through it together.

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And I'll give you a little bit of a hint.

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Let's imagine that this
rectangle represents a whole

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and notice it is divided
into six equal sections

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so each of those sections is 1/6.

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And so if I start at zero,
how many would I fill in

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to get 2/6?

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And what would be the corresponding point

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on the number line?

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All right, now let's do it together.

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So if each of these is a sixth,

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and we have 6/6 there
so that would be a whole

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and that's good because
it goes from zero to one

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and you can view one as a whole.

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2/6 is, so that's 1/6 right over there

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and then that is 2/6.

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And so you can see on the number line,

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the thing that gets us 2/6 of
the way to one is at point B.

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It corresponds to how much
we've filled up that rectangle,

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point B right over there.

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Now another way that you
could think about it,

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you could see that the
space between zero and one

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is split up into six equal sections.

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One, two, three, four,
five, six equal sections.

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And we want to go to 2/6.

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2/6.

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So each of those equal sections,

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we are increasing by a sixth.

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So we're going from zero to 1/6 to 2/6.

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Once again we end up at point B.


Graphing logarithmic functions (example 2)

WEBVTT
Kind: captions
Language: en

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- [Instructor] This is a screenshot

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from an exercise on Khan Academy,

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and it says the intergraphic,
the interactive graph below

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contains the graph of y is equal

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to log base two of x as a dashed curve,

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and you can see it down
there as a dashed curve,

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with the points one comma zero

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and two comma one highlighted.

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Adjust the movable
graph to draw y is equal

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to four times log base two
of x plus six minus seven.

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And so if you happen to have
this exercise in front of you

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I encourage you to do that.

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Or if you're just thinking
about in your head,

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think about how you would approach this.

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And I'll give you a hint,
to go from our original

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y is equal to log base
two of x to all of this,

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it's really going to be a
series of transformations.

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And on this tool right over here,

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what we can do is we can move
this vertical asymptote around

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so that's one thing we can move,

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and then we can also
move two of these points.

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So where we're starting is right,

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we are starting right over there.

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And so let's see, and
that was just the graph

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of y is equal to log base two of x.

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So let's just do these
transformations one at a time.

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So the first thing I am going to do,

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instead of just doing log base two of x,

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let's do log base two of x plus six.

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So if you replace your
x with an x plus six,

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what is it going to do?

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Well it's going to shift
everything six to the left,

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and if that doesn't make
intuitive sense to you,

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I encourage you to watch some

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of the introductory videos
on shifting transformations.

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So everything is going
to shift six to the left.

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So this vertical asymptote is
going to shift six to the left

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it's gonna be, instead of
being at x equals zero,

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it's going to go all the way
to x equals negative six.

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This point right over here,
which was at one comma zero,

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it's going to go six to the left,

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one, two, three, four, five, six.

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And this point, which as at two comma one,

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is gonna go six to the left,

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one, two, three, four, five, and six.

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So so far what we have graphed

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is log base two of x plus six.

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So the next thing we might wanna do is

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what is four time log
base two of x plus six.

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And I want you to think about it is

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whatever y-value we were getting before,

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we're now going to get four times that.

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So when x is equal to negative five,

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we're getting a y-value of zero,

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but four times zero is still zero,

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so that point will stay the same.

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But when x is equal to negative four,

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we're getting a y-value of one,

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but now that's going to
be four times higher,

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'cause we're putting that four out front,

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so instead of being at four,

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instead of being at one
it's going to be at four.

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So this right over here is the graph

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of y is equal to log
base two of x plus six.

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And then the last thing
we have to consider

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is well we're gonna take all of that

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and then we're going to subtract seven

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to get to our target graph.

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So whatever points we are here,

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we are now going to subtract seven.

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So this is at y equals zero,

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but now we're going to subtract
seven, so we're going to

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go down one, two, three,
four, five, six, seven,

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I went off the screen a
little bit, but let me see

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if I can scroll down a little
bit so that you can see that,

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almost, there you go, now you can see.

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I moved this down from
zero to negative seven,

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and then this one I
have to move down seven,

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one, two, three, four,
five, six, and seven,

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and we're done, there you have it.

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That is the graph of y is equal
to four times log base two

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of x plus six minus
seven, and we are done.


Graphing logarithmic functions (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.070
- [Instructor] We are told
the graph of y is equal

00:00:03.070 --> 00:00:05.850
to log base two of x is shown below,

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and they say graph y is equal

00:00:07.630 --> 00:00:12.340
to two log base two of
negative x minus three.

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So pause this video and have a go at it.

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The way to think about it
is that this second equation

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that we wanna graph is really
based on this first equation

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through a series of transformations.

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So I encourage you to
take some graph paper out

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and sketch how those transformations

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would affect our original graph

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to get to where we need to go.

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All right, now let's do this together.

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So what we already have graphed,

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I'll just write it in purple,

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is y is equal to log base two of x.

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Now the difference between
what I just wrote in purple

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and where we wanna go is in the first case

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we don't multiply anything
times our log base two of x,

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while in our end goal we multiply by two.

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In our first situation, we
just have log base two of x

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while in here we have log base two

00:00:58.530 --> 00:01:01.947
of negative x minus three.

00:01:01.947 --> 00:01:03.830
And in fact we could even view that

00:01:03.830 --> 00:01:08.830
as it's the negative of x plus three.

00:01:10.310 --> 00:01:13.590
So what we could do is try to
keep changing this equation

00:01:13.590 --> 00:01:15.720
and that's going to transform its graph

00:01:15.720 --> 00:01:18.220
until we get to our goal.

00:01:18.220 --> 00:01:20.580
So maybe the first thing
we might want to do

00:01:20.580 --> 00:01:23.950
is let's replace our x with a negative x.

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So let's try to graph y is equal

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to log base two of negative x.

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In other videos we've talked about

00:01:30.620 --> 00:01:32.600
what transformation would go on there,

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but we can intuit through it as well.

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Now whatever value y would have taken on

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at a given x-value, so for
example when x equals four

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log base two of four is two,

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now that will happen at negative four.

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So log base two of the
negative of negative four,

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well that's still log base two of four,

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so that's still going to be two.

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And if you were to put in let's say a,

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whatever was happening at one before,

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log base two of one is zero,

00:02:01.800 --> 00:02:03.270
but now that's going to
happen at negative one

00:02:03.270 --> 00:02:05.410
'cause you take the
negative of negative one,

00:02:05.410 --> 00:02:07.430
you're gonna get a one over here,

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so log base two of one is zero.

00:02:11.190 --> 00:02:14.570
And so similarly when
you had at x equals eight

00:02:14.570 --> 00:02:16.040
you got to three, now
that's going to happen

00:02:16.040 --> 00:02:20.000
at x equals negative eight
we are going to be at three.

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And so the graph is going
to look something like

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what I am graphing right over here.

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All right, fair enough.

00:02:32.750 --> 00:02:35.020
Now the next thing we might wanna do is

00:02:35.020 --> 00:02:39.970
hey let's replace this
x with an x plus three,

00:02:39.970 --> 00:02:42.090
'cause that'll get us at least,

00:02:42.090 --> 00:02:43.570
in terms of what we're taking the log of,

00:02:43.570 --> 00:02:45.930
pretty close to our original equation.

00:02:45.930 --> 00:02:50.840
So now let's think about y
is equal to log base two of,

00:02:50.840 --> 00:02:53.000
and actually I should put
parentheses in that previous one

00:02:53.000 --> 00:02:54.350
just so it's clear,

00:02:54.350 --> 00:02:58.760
so log base two of not
just the negative of x,

00:02:58.760 --> 00:03:03.360
but we're going to replace
x with x plus three.

00:03:03.360 --> 00:03:06.690
Now what happens if you
replace x with an x plus three?

00:03:06.690 --> 00:03:08.960
Or you could even view x
plus three as the same thing

00:03:08.960 --> 00:03:11.870
as x minus negative three.

00:03:11.870 --> 00:03:13.800
Well we've seen in multiple examples

00:03:13.800 --> 00:03:15.680
that when you replace
x with an x plus three

00:03:15.680 --> 00:03:19.890
that will shift your entire
graph three to the left.

00:03:19.890 --> 00:03:24.890
So this shifts, shifts three to the left.

00:03:27.100 --> 00:03:29.630
If it was an x minus three in here,

00:03:29.630 --> 00:03:30.850
you would should three to the right.

00:03:30.850 --> 00:03:33.150
So how do we shift three to the left?

00:03:33.150 --> 00:03:36.390
Well when the point
where we used to hit zero

00:03:36.390 --> 00:03:38.410
are now going to happen
three to the left of that.

00:03:38.410 --> 00:03:41.170
So we used to hit it at
x equals negative one,

00:03:41.170 --> 00:03:43.720
now it's going to happen
at x equals negative four.

00:03:43.720 --> 00:03:45.810
The point at which y is equal to two,

00:03:45.810 --> 00:03:47.460
instead of happening at
x equals negative four,

00:03:47.460 --> 00:03:49.240
is now going to happen
three to the left of that

00:03:49.240 --> 00:03:52.130
which is x equals negative seven,

00:03:52.130 --> 00:03:56.383
so it's going to be right over there.

00:03:56.383 --> 00:04:00.560
And the point at which the
graph goes down to infinity,

00:04:00.560 --> 00:04:03.330
that was happening as x approaches zero,

00:04:03.330 --> 00:04:05.060
now that's going to happen

00:04:05.060 --> 00:04:07.760
as x approaches three to the left of that,

00:04:07.760 --> 00:04:09.620
as x approaches negative three,

00:04:09.620 --> 00:04:12.900
so I could draw a little
dotted line right over here

00:04:12.900 --> 00:04:15.620
to show that as x approaches that

00:04:15.620 --> 00:04:16.800
our graph is going to approach zero.

00:04:16.800 --> 00:04:21.080
So our graph's gonna look
something, something like this,

00:04:23.418 --> 00:04:26.160
like this, this is all hand-drawn

00:04:26.160 --> 00:04:29.210
so it's not perfectly drawn
but we're awfully close.

00:04:29.210 --> 00:04:31.670
Now to get from where we are to our goal,

00:04:31.670 --> 00:04:34.640
we just have to multiply
the right hand side by two.

00:04:34.640 --> 00:04:39.640
So now let's graph y, not
two, let's graph y is equal

00:04:40.190 --> 00:04:45.190
to two log base two of
negative of x plus three,

00:04:46.770 --> 00:04:49.000
which is the exact same
goal as we had before,

00:04:49.000 --> 00:04:50.840
I've just factored out the negative

00:04:50.840 --> 00:04:52.640
to help with our transformations.

00:04:52.640 --> 00:04:55.450
So all that means is whatever
y value we were taking on

00:04:55.450 --> 00:04:59.180
at a given x you're now going
to take on twice that y-value.

00:04:59.180 --> 00:05:02.540
So where you were at zero,
you're still going to be zero.

00:05:02.540 --> 00:05:03.700
But where you were two,

00:05:03.700 --> 00:05:06.420
you are now going to be equal to four,

00:05:06.420 --> 00:05:09.540
and so the graph is
going to look something,

00:05:09.540 --> 00:05:14.540
something like what I
am drawing right now.

00:05:15.490 --> 00:05:17.930
And we're done, that's
our sketch of the graph

00:05:17.930 --> 00:05:20.440
of all of this business.

00:05:20.440 --> 00:05:21.760
And once again, if you're
doing it on Khan Academy,

00:05:21.760 --> 00:05:23.380
there would be a choice
that looks like this

00:05:23.380 --> 00:05:25.713
and you would hopefully pick that one.


Dividing polynomials by x (no remainders)

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.370
- [Instructor] What I'd
like to do in this video

00:00:02.370 --> 00:00:05.250
is try to figure out what x to the fourth

00:00:05.250 --> 00:00:10.050
minus two x to the third plus five x

00:00:10.050 --> 00:00:14.610
divided by x is equal to.

00:00:14.610 --> 00:00:16.820
So pause this video and see
if you can have a go at that

00:00:16.820 --> 00:00:18.670
before we work through this together.

00:00:19.550 --> 00:00:23.000
All right, so if we're saying
what is this top expression

00:00:23.000 --> 00:00:24.810
divided by this bottom expression,

00:00:24.810 --> 00:00:26.930
another way to think about it is,

00:00:26.930 --> 00:00:29.500
what do I have to multiply,
so I'm going to multiply

00:00:29.500 --> 00:00:31.410
something, I'll put that in parentheses.

00:00:31.410 --> 00:00:34.300
If I multiply that something times x,

00:00:34.300 --> 00:00:36.930
I should get x to the fourth

00:00:36.930 --> 00:00:40.670
minus two x to the third plus five x.

00:00:40.670 --> 00:00:42.590
Now how do I approach that?

00:00:42.590 --> 00:00:45.520
Well there's two ways
that I could tackle it.

00:00:45.520 --> 00:00:49.650
One way is I could just rewrite
this expression as being,

00:00:49.650 --> 00:00:51.780
and I will just make this x in yellow

00:00:51.780 --> 00:00:53.140
so I can keep track of it.

00:00:53.140 --> 00:00:58.140
I could just rewrite
this as one over x times,

00:00:58.150 --> 00:01:00.190
times x to the fourth

00:01:00.190 --> 00:01:04.090
minus two x to the third plus five x.

00:01:04.090 --> 00:01:07.410
And then I can distribute the one over x,

00:01:07.410 --> 00:01:10.070
and so what is that going to be equal to?

00:01:10.070 --> 00:01:13.820
Well it's going to be
equal to x to the fourth.

00:01:13.820 --> 00:01:18.300
Let me do this, x to the fourth over x

00:01:18.300 --> 00:01:22.553
minus two x to the third over x

00:01:24.730 --> 00:01:29.327
plus five x, plus five x over x.

00:01:31.330 --> 00:01:34.130
And so what are each of
these going to be equal to?

00:01:34.130 --> 00:01:36.280
X to the fourth divided by x,

00:01:36.280 --> 00:01:39.250
if I have four x's that
I'm multiplying together

00:01:39.250 --> 00:01:41.860
and then I divide by x,
that's going to be equivalent

00:01:41.860 --> 00:01:43.730
to x to the third power.

00:01:43.730 --> 00:01:47.730
So this right over here is
equal to x to the third.

00:01:47.730 --> 00:01:49.930
You could also get there from
your exponent properties.

00:01:49.930 --> 00:01:52.380
In the denominator, you have
an x to the first power,

00:01:52.380 --> 00:01:53.990
and so you would subtract the exponents.

00:01:53.990 --> 00:01:56.570
You have the same base here,
so that's x to the third.

00:01:56.570 --> 00:01:59.250
And then, in this part right over here,

00:01:59.250 --> 00:02:00.940
what would that equal to?

00:02:00.940 --> 00:02:02.430
Well it's going to be minus two

00:02:02.430 --> 00:02:05.450
x to the third divided by x to the first.

00:02:05.450 --> 00:02:08.760
Well by the same property,
that's going to be x squared.

00:02:08.760 --> 00:02:12.080
And then last but not
least, if you take five x's

00:02:12.080 --> 00:02:13.700
and then you divide by x,

00:02:13.700 --> 00:02:16.420
you are just going to be left with five.

00:02:16.420 --> 00:02:19.330
And you can verify that this, indeed,

00:02:19.330 --> 00:02:20.960
if I were to multiply it by x,

00:02:20.960 --> 00:02:22.680
I'm gonna get x to the fourth minus two x

00:02:22.680 --> 00:02:24.800
to the third plus five x.

00:02:24.800 --> 00:02:25.710
Let me do that.

00:02:25.710 --> 00:02:29.000
If I put x to the third
minus two x squared

00:02:29.000 --> 00:02:32.230
plus five times x, what I
could do is distribute the x.

00:02:32.230 --> 00:02:34.670
X times x to the third is x to the fourth.

00:02:34.670 --> 00:02:36.310
X times negative two x squared

00:02:36.310 --> 00:02:38.260
is negative two x to the third.

00:02:38.260 --> 00:02:41.340
X times five is five x.

00:02:41.340 --> 00:02:43.440
Now I mentioned there's two
ways that I could do it.

00:02:43.440 --> 00:02:45.650
Another way that I could
try to tackle it is

00:02:45.650 --> 00:02:46.980
I could look at this numerator

00:02:46.980 --> 00:02:48.610
and try to factor an x out.

00:02:48.610 --> 00:02:51.220
I would try to factor out
whatever I see in the denominator.

00:02:51.220 --> 00:02:54.590
So if I do that, actually let
me just rewrite the numerator.

00:02:54.590 --> 00:02:58.470
So I can rewrite x to the fourth

00:02:58.470 --> 00:03:01.950
as x times x to the third.

00:03:01.950 --> 00:03:06.290
And then I can rewrite the
minus two x to the third

00:03:06.290 --> 00:03:08.390
as, let me write it this way,

00:03:08.390 --> 00:03:13.390
as plus x times negative two x squared.

00:03:14.410 --> 00:03:18.560
And then I could write this five x

00:03:18.560 --> 00:03:22.800
as being equal to plus x times five.

00:03:22.800 --> 00:03:25.730
And then I'm gonna divide everything by x,

00:03:25.730 --> 00:03:26.700
divide everything by x.

00:03:26.700 --> 00:03:28.640
I just rewrote the numerator here,

00:03:28.640 --> 00:03:32.150
but for each of those
terms, I factored out an x.

00:03:32.150 --> 00:03:35.300
And now I can factor out
x out of the whole thing.

00:03:35.300 --> 00:03:38.150
So I sometimes think of factoring out an x

00:03:38.150 --> 00:03:40.620
out of the whole thing as
reverse distributive property.

00:03:40.620 --> 00:03:44.010
So if I factor out this
x out of every term,

00:03:44.010 --> 00:03:44.843
what am I left with?

00:03:44.843 --> 00:03:47.140
I'm left with an x times

00:03:48.010 --> 00:03:53.010
x to the third minus two x squared

00:03:53.380 --> 00:03:57.700
plus five; I ended up doing
that in the same color,

00:03:57.700 --> 00:04:00.860
but hopefully you're following, plus five,

00:04:00.860 --> 00:04:03.070
and then all of that is divided by x.

00:04:03.070 --> 00:04:05.300
And as long as x does not equal zero,

00:04:05.300 --> 00:04:08.090
x divided by x is going
to be equal to one,

00:04:08.090 --> 00:04:10.310
and we're left with what
we had to begin with,

00:04:10.310 --> 00:04:12.910
or the answer that we had to begin with.

00:04:12.910 --> 00:04:14.740
So these are two different approaches.

00:04:14.740 --> 00:04:17.760
Nothing super sophisticated here.

00:04:17.760 --> 00:04:19.860
When you're dividing by
x, you're just like hey,

00:04:19.860 --> 00:04:21.310
that's the same thing as
multiplying every term

00:04:21.310 --> 00:04:24.160
by one over x, or you can factor out an x

00:04:24.160 --> 00:04:27.503
out of the numerator,
and then they cancel out.


Multiplying monomials

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.910
- [Instructor] Let's say
that we wanted to multiply

00:00:02.910 --> 00:00:07.410
five x squared and,
I'll do this in purple,

00:00:07.410 --> 00:00:11.260
three x to the fifth,
what would this equal?

00:00:11.260 --> 00:00:12.940
Pause this video and see if you can

00:00:12.940 --> 00:00:15.160
reason through that a little bit.

00:00:15.160 --> 00:00:16.980
All right, now let's work
through this together.

00:00:16.980 --> 00:00:18.660
And really, all we're going to do

00:00:18.660 --> 00:00:20.710
is use properties of multiplication

00:00:20.710 --> 00:00:22.550
and use properties of exponents

00:00:22.550 --> 00:00:25.040
to essentially rewrite this expression.

00:00:25.040 --> 00:00:26.640
So we can just view this,

00:00:26.640 --> 00:00:28.780
if we're just multiplying
a bunch of things,

00:00:28.780 --> 00:00:31.230
it doesn't matter what
order we multiply them in.

00:00:31.230 --> 00:00:33.540
So you can just view this
as five times x squared

00:00:33.540 --> 00:00:35.830
times three times x to the fifth,

00:00:35.830 --> 00:00:39.760
or we could multiply our
five and three first,

00:00:39.760 --> 00:00:44.760
so you could view this as
five times three, times three,

00:00:44.960 --> 00:00:48.720
times x squared, times x squared,

00:00:48.720 --> 00:00:52.890
times x to the fifth,
times x to the fifth.

00:00:52.890 --> 00:00:55.870
And now what is five times three?

00:00:55.870 --> 00:00:58.400
I think you know that, that is 15.

00:00:58.400 --> 00:01:02.669
Now what is x squared
times x to the fifth?

00:01:02.669 --> 00:01:04.472
Now some of you might recognize

00:01:04.472 --> 00:01:06.970
that exponent properties
would come into play here.

00:01:06.970 --> 00:01:09.290
If I'm multiplying two things like this,

00:01:09.290 --> 00:01:11.470
so we have the some base
and different exponents,

00:01:11.470 --> 00:01:13.960
that this is going to
be equal to x to the,

00:01:13.960 --> 00:01:16.100
and we add these two exponents,

00:01:16.100 --> 00:01:20.030
x to the two plus five power,
or x to the seventh power.

00:01:20.030 --> 00:01:22.240
If what I just did seems
counterintuitive to you

00:01:22.240 --> 00:01:24.500
I'll just remind you, what is x squared?

00:01:24.500 --> 00:01:27.030
x squared is x times x.

00:01:27.030 --> 00:01:28.840
And what is x to the fifth?

00:01:28.840 --> 00:01:33.840
That is x times x times x times x times x.

00:01:34.520 --> 00:01:38.360
And if you multiply them all
together what do you get?

00:01:38.360 --> 00:01:39.760
Well you got seven x's

00:01:39.760 --> 00:01:41.380
and you multiply them all together

00:01:41.380 --> 00:01:43.600
and that is x to the seventh.

00:01:43.600 --> 00:01:44.740
And so there you have it,

00:01:44.740 --> 00:01:46.550
five x squared times three x to the fifth

00:01:46.550 --> 00:01:49.840
is 15x to the seventh power.

00:01:49.840 --> 00:01:52.650
So the key is, is look
at these coefficients,

00:01:52.650 --> 00:01:55.680
look at these numbers, a five
and a three, multiply those,

00:01:55.680 --> 00:01:57.880
and then for any variable you have,

00:01:57.880 --> 00:02:00.460
you have x here, so
you have a common base,

00:02:00.460 --> 00:02:03.220
then you can add those exponents,

00:02:03.220 --> 00:02:06.770
and what we just did is known
as multiplying monomials,

00:02:06.770 --> 00:02:11.770
which sounds very fancy, but
this is a monomial, monomial,

00:02:12.440 --> 00:02:13.880
and in the future we'll do

00:02:13.880 --> 00:02:15.720
multiplying things like polynomials

00:02:15.720 --> 00:02:18.840
where we have multiple of
these things added together.

00:02:18.840 --> 00:02:21.610
But that's all it is,
multiplying monomials.

00:02:21.610 --> 00:02:22.860
Let's do one more example,

00:02:22.860 --> 00:02:24.810
and let's use a different
variable this time,

00:02:24.810 --> 00:02:27.370
just to get some variety in there.

00:02:27.370 --> 00:02:29.468
Let's say we wanna multiply the monomial

00:02:29.468 --> 00:02:32.540
three t to the seventh power,

00:02:32.540 --> 00:02:37.310
times another monomial negative four t.

00:02:37.310 --> 00:02:39.984
Pause this video and see if
you can work through that.

00:02:39.984 --> 00:02:43.480
All right, so I'm gonna do
this one a little bit faster.

00:02:43.480 --> 00:02:46.750
I am going to look at the
three and the negative four

00:02:46.750 --> 00:02:48.590
and I'm gonna multiply those first,

00:02:48.590 --> 00:02:50.750
and I'm going to get a negative 12.

00:02:50.750 --> 00:02:53.930
And then if I were to want to multiply

00:02:53.930 --> 00:02:56.960
the t to the seventh times t,

00:02:56.960 --> 00:03:00.290
once again they're both
the variable t as our base,

00:03:00.290 --> 00:03:02.250
so that's going to be t to the seventh

00:03:02.250 --> 00:03:05.290
times t to the first
power, that's what t is,

00:03:05.290 --> 00:03:08.590
that's going to be t to
the seven plus one power,

00:03:08.590 --> 00:03:10.030
or t to the eighth.

00:03:10.030 --> 00:03:12.170
But there you go, we are done again,

00:03:12.170 --> 00:03:14.993
we just multiplied
another set of monomials.


Graphing square and cube root functions

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.920
- [Instructor] We're told
the graph of y is equal

00:00:01.920 --> 00:00:04.980
to square root of x is
shown below, fair enough.

00:00:04.980 --> 00:00:06.560
Which of the following is the graph

00:00:06.560 --> 00:00:10.360
of y is equal to two times the square root

00:00:10.360 --> 00:00:13.360
of negative x minus one?

00:00:13.360 --> 00:00:15.980
And they give us some choices here,

00:00:15.980 --> 00:00:19.140
and so I encourage you to pause this video

00:00:19.140 --> 00:00:20.440
and try to figure it out on your own

00:00:20.440 --> 00:00:22.423
before we work through this together.

00:00:23.360 --> 00:00:24.690
All right, now let's work
through this together,

00:00:24.690 --> 00:00:25.700
and the way that I'm going to do it

00:00:25.700 --> 00:00:27.430
is I'm actually going to try to draw

00:00:27.430 --> 00:00:29.200
what the graph of two
times the square root

00:00:29.200 --> 00:00:31.370
of negative x minus one should look like,

00:00:31.370 --> 00:00:33.300
and then I'll just look
at which of the choices

00:00:33.300 --> 00:00:34.990
is closest to what I drew.

00:00:34.990 --> 00:00:36.270
And the way that I'm going to do that

00:00:36.270 --> 00:00:37.740
is I'm going to do it step by step,

00:00:37.740 --> 00:00:41.260
so we already see what
y equals the square root

00:00:41.260 --> 00:00:42.900
of x looks like,

00:00:42.900 --> 00:00:44.930
but let's say we just want to build up.

00:00:44.930 --> 00:00:47.040
So let's say we want to now figure out

00:00:47.040 --> 00:00:50.160
what is the graph of y is
equal to the square root of?

00:00:50.160 --> 00:00:52.170
Instead of an x under the radical sign,

00:00:52.170 --> 00:00:55.210
let me put a negative x
under the radical sign.

00:00:55.210 --> 00:00:56.980
What would that do to it?

00:00:56.980 --> 00:00:59.870
Well, whatever was happening
at a certain value of x

00:00:59.870 --> 00:01:02.540
will now happen at the
negative of that value of x.

00:01:02.540 --> 00:01:05.220
So the square root of x is not
defined for negative numbers.

00:01:05.220 --> 00:01:08.100
Now this one won't be
defined for positive numbers.

00:01:08.100 --> 00:01:10.690
And the behavior that
you saw at x equals two,

00:01:10.690 --> 00:01:13.710
you would now see at
x equals negative two.

00:01:13.710 --> 00:01:15.470
The behavior that you
saw at x equals four,

00:01:15.470 --> 00:01:18.660
you will now see at x
equals negative four,

00:01:18.660 --> 00:01:21.110
and so on and so forth.

00:01:21.110 --> 00:01:24.350
So the y equals the
square root of negative x

00:01:25.327 --> 00:01:26.970
is going to look like this.

00:01:26.970 --> 00:01:30.200
You've essentially flipped it over the y.

00:01:30.200 --> 00:01:32.770
We have flipped it over the y axis.

00:01:32.770 --> 00:01:35.090
All right, so we've done this part.

00:01:35.090 --> 00:01:36.270
Now let's scale that.

00:01:36.270 --> 00:01:38.820
Now let's multiply that by two.

00:01:38.820 --> 00:01:41.840
So what would y is equal to
two times the square root

00:01:41.840 --> 00:01:43.453
of negative x look like?

00:01:44.340 --> 00:01:45.950
Well, it would look like this red curve,

00:01:45.950 --> 00:01:47.140
but at any given x value,

00:01:47.140 --> 00:01:49.220
we're gonna get twice as high.

00:01:49.220 --> 00:01:51.480
So at x equals negative four,

00:01:51.480 --> 00:01:53.210
instead of getting to two,

00:01:53.210 --> 00:01:54.810
we're now going to get to four.

00:01:54.810 --> 00:01:56.520
At x equals negative nine,

00:01:56.520 --> 00:01:58.210
instead of getting to three,

00:01:58.210 --> 00:02:00.240
we are now going to get to six.

00:02:00.240 --> 00:02:02.340
Now at x equals zero, we're
still going to be at zero

00:02:02.340 --> 00:02:03.940
'cause two times zero is zero,

00:02:03.940 --> 00:02:05.420
so it's going to look,

00:02:05.420 --> 00:02:08.040
it's going to look like that.

00:02:08.040 --> 00:02:09.980
Something like that,

00:02:09.980 --> 00:02:12.890
so that's y equals two times
the square root of negative x.

00:02:12.890 --> 00:02:15.300
And then last but not least,

00:02:15.300 --> 00:02:17.630
what will y,

00:02:17.630 --> 00:02:19.430
let me do that in a different color.

00:02:19.430 --> 00:02:23.810
What will y equals two
times the square root

00:02:23.810 --> 00:02:27.030
of negative x minus one look like?

00:02:27.030 --> 00:02:28.930
Well, whatever y value
we were getting before,

00:02:28.930 --> 00:02:32.760
we're now just going to
shift everything down by one.

00:02:32.760 --> 00:02:34.020
So if we were at six before,

00:02:34.020 --> 00:02:35.613
we're going to be at five now.

00:02:36.989 --> 00:02:38.070
If we were at four before,

00:02:38.070 --> 00:02:39.980
we're now going to be at three.

00:02:39.980 --> 00:02:41.570
If we were at zero before,

00:02:41.570 --> 00:02:44.140
we're now going to be at negative one,

00:02:44.140 --> 00:02:48.390
and so our curve is going
to look something like,

00:02:48.390 --> 00:02:50.500
something like that.

00:02:50.500 --> 00:02:54.920
So let's look for, let's see
which choices match that.

00:02:54.920 --> 00:02:59.000
So let me scroll down here,

00:02:59.000 --> 00:03:03.320
and both C and D kind of
look right, but notice,

00:03:03.320 --> 00:03:05.740
right at zero, we want
it to be at negative one,

00:03:05.740 --> 00:03:10.347
so D is exactly what we had drawn.

00:03:10.347 --> 00:03:11.990
And at nine, we're at five.

00:03:11.990 --> 00:03:14.100
Or at negative nine, we're at five.

00:03:14.100 --> 00:03:15.730
At negative four, we're at three,

00:03:15.730 --> 00:03:17.010
and at zero, we're at negative one.

00:03:17.010 --> 00:03:18.720
Exactly what we had drawn.

00:03:18.720 --> 00:03:19.973
Let's do another example.

00:03:20.860 --> 00:03:21.910
So here,

00:03:21.910 --> 00:03:24.880
this is a similar question.

00:03:24.880 --> 00:03:26.850
Now they graphed the cube root of x.

00:03:26.850 --> 00:03:28.380
Y is equal to the cube root of x,

00:03:28.380 --> 00:03:29.730
and then they say which of the following

00:03:29.730 --> 00:03:31.780
is the graph of this business?

00:03:31.780 --> 00:03:34.530
And they give us choices again,

00:03:34.530 --> 00:03:36.880
so once again, pause this
video and try to work it out

00:03:36.880 --> 00:03:39.193
on your own before we do this together.

00:03:40.040 --> 00:03:42.700
All right, now let's work on this together

00:03:42.700 --> 00:03:44.050
and I'm gonna do the same technique.

00:03:44.050 --> 00:03:46.860
I'm just gonna build it up piece by piece.

00:03:46.860 --> 00:03:51.860
So this is already y is
equal to the cube root of x.

00:03:53.000 --> 00:03:54.810
So now let's build up on that.

00:03:54.810 --> 00:03:57.990
Let's say we want to
now have an x plus two

00:03:57.990 --> 00:03:59.860
under the radical sign.

00:03:59.860 --> 00:04:02.380
So let's graph y is equal to the cube root

00:04:02.380 --> 00:04:05.430
of x plus two.

00:04:05.430 --> 00:04:10.430
Well, what this does is it
shifts the curve two to the left.

00:04:11.080 --> 00:04:14.650
And we've gone over this
in multiple videos before,

00:04:14.650 --> 00:04:16.210
so we are now here,

00:04:16.210 --> 00:04:18.190
and you could even try some values out

00:04:18.190 --> 00:04:19.700
to verify that.

00:04:19.700 --> 00:04:23.930
At x equals zero,

00:04:23.930 --> 00:04:26.410
at x equals zero, or actually,
let me put it this way.

00:04:26.410 --> 00:04:27.780
At x equals negative two,

00:04:27.780 --> 00:04:30.610
you're gonna kick the cube root of zero,

00:04:30.610 --> 00:04:32.010
which is right over there.

00:04:32.010 --> 00:04:35.320
So we have now shifted two to the left

00:04:35.320 --> 00:04:37.030
to look something,

00:04:37.030 --> 00:04:39.730
to look something like this,

00:04:39.730 --> 00:04:42.970
and now, let's build up on that.

00:04:42.970 --> 00:04:44.910
Let's multiply this times a negative,

00:04:44.910 --> 00:04:47.470
so y is equal to the
negative of the cube root

00:04:47.470 --> 00:04:49.180
of x plus two.

00:04:49.180 --> 00:04:50.550
What would that look like?

00:04:50.550 --> 00:04:52.740
Well if you multiply
your whole expression,

00:04:52.740 --> 00:04:56.040
or in this case, the whole
graph or the whole function

00:04:56.040 --> 00:04:56.873
by a negative,

00:04:56.873 --> 00:05:00.230
you're gonna flip it
over the horizontal axis.

00:05:00.230 --> 00:05:02.880
And so it is now going to look like this.

00:05:02.880 --> 00:05:04.470
Whatever y value we're gonna get before

00:05:04.470 --> 00:05:06.510
for a given x, you're
now getting the opposite,

00:05:06.510 --> 00:05:07.800
the negative of it.

00:05:07.800 --> 00:05:09.410
So it's going to look,

00:05:09.410 --> 00:05:12.270
it's going to look like
that, something like that.

00:05:12.270 --> 00:05:14.720
So that is y equal to the
negative of the cube root

00:05:14.720 --> 00:05:16.270
of x plus two.

00:05:16.270 --> 00:05:19.110
And then last, but not least,

00:05:19.110 --> 00:05:21.400
we are going to think about,

00:05:21.400 --> 00:05:23.400
and I'm searching for
an appropriate color.

00:05:23.400 --> 00:05:24.910
I haven't used orange yet.

00:05:24.910 --> 00:05:28.820
Y is equal to the negative of
the cube root of x plus two,

00:05:28.820 --> 00:05:30.390
and I'm going to add five.

00:05:30.390 --> 00:05:32.280
So all that's going to do
is take this last graph

00:05:32.280 --> 00:05:33.900
and shift it up by five.

00:05:33.900 --> 00:05:35.380
Whatever y value I was
going to get before,

00:05:35.380 --> 00:05:37.240
now I'm going to get five higher.

00:05:37.240 --> 00:05:39.260
So five higher, let's see.

00:05:39.260 --> 00:05:40.320
I was at zero here,

00:05:40.320 --> 00:05:43.360
so I'm now going to be at five here.

00:05:43.360 --> 00:05:44.650
So that's going to look,

00:05:44.650 --> 00:05:46.440
it's going to look something,

00:05:46.440 --> 00:05:48.530
something like,

00:05:48.530 --> 00:05:51.530
something like that.

00:05:51.530 --> 00:05:53.490
And I'm not drawing it perfectly,

00:05:53.490 --> 00:05:57.160
but you get the general, the general idea,

00:05:57.160 --> 00:05:58.100
now let's look at the choices.

00:05:58.100 --> 00:05:59.730
And I think the key point to look at

00:05:59.730 --> 00:06:01.820
is this point right over here,

00:06:01.820 --> 00:06:04.050
that in our original
graph, was at zero, zero.

00:06:04.050 --> 00:06:08.460
Now it is going to be at
negative two, comma, five.

00:06:08.460 --> 00:06:10.760
So let's look for it,

00:06:10.760 --> 00:06:12.160
and it also should be flipped.

00:06:12.160 --> 00:06:14.150
So on the left hand side,
we have the top part

00:06:14.150 --> 00:06:15.090
and on the right hand side,

00:06:15.090 --> 00:06:17.570
we have the part that goes lower.

00:06:17.570 --> 00:06:18.643
So let's see.

00:06:19.600 --> 00:06:23.840
So A, C, and B all have the left hand side

00:06:23.840 --> 00:06:24.900
as the higher part

00:06:24.900 --> 00:06:27.040
and then the right hand
side being the lower part,

00:06:27.040 --> 00:06:30.190
but we wanted this point to be
at negative two, comma, five.

00:06:30.190 --> 00:06:31.960
A doesn't have it there.

00:06:31.960 --> 00:06:33.870
B doesn't have it there.

00:06:33.870 --> 00:06:35.690
D we already said goes
to the wrong direction.

00:06:35.690 --> 00:06:36.523
It's increasing.

00:06:38.395 --> 00:06:39.228
So let's see,

00:06:39.228 --> 00:06:40.620
negative two, comma, five,

00:06:40.620 --> 00:06:42.520
it's indeed what we expected.

00:06:42.520 --> 00:06:45.810
This is pretty close to what
we had drawn on our own,

00:06:45.810 --> 00:06:47.033
so choice C.


Scaling functions horizontally: examples

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.500
- [Instructor] We're told this
is the graph of function f,

00:00:02.500 --> 00:00:03.380
fair enough.

00:00:03.380 --> 00:00:07.270
Function g is defined as g
of x is equal to f of two x.

00:00:07.270 --> 00:00:08.690
What is the graph of g?

00:00:08.690 --> 00:00:12.120
So pause this video, and try
to figure that out on your own.

00:00:12.120 --> 00:00:13.490
All right, now let's work through this.

00:00:13.490 --> 00:00:15.390
And the way I will think about it,

00:00:15.390 --> 00:00:17.240
I'll set up a little table here.

00:00:17.240 --> 00:00:18.933
And I'll have an x column,

00:00:21.089 --> 00:00:24.680
and then I'll have a, well,
actually just put g of x column.

00:00:24.680 --> 00:00:28.370
And of course, g of x
is equal to f of two x.

00:00:28.370 --> 00:00:31.210
So when x is,

00:00:31.210 --> 00:00:33.520
and actually let me
see, when x is equal to,

00:00:33.520 --> 00:00:36.540
I could pick a point like x equaling zero,

00:00:36.540 --> 00:00:41.420
so g of zero is going to
be f of two times zero.

00:00:41.420 --> 00:00:43.560
So it's going to be f of two times zero,

00:00:43.560 --> 00:00:45.720
which is still f of zero,

00:00:45.720 --> 00:00:49.490
which is going to be equal
to a little bit over four,

00:00:49.490 --> 00:00:53.180
so which is equal to f of zero.

00:00:53.180 --> 00:00:57.930
And so they're going to both
have the same y-intercept,

00:00:57.930 --> 00:01:00.110
but interesting things are
going to happen the further

00:01:00.110 --> 00:01:02.090
that we get from the y-axis

00:01:02.090 --> 00:01:05.160
or as our x increases in
either direction away,

00:01:05.160 --> 00:01:08.720
or as our x gets bigger in
either direction from zero.

00:01:08.720 --> 00:01:11.160
So let's think about
what's going to happen

00:01:11.160 --> 00:01:13.770
at x equals two.

00:01:13.770 --> 00:01:15.120
So at x equals two,

00:01:15.120 --> 00:01:19.490
g of two is going to be
equal to f of two times two,

00:01:19.490 --> 00:01:22.120
two times two, which
is equal to f of four.

00:01:22.120 --> 00:01:24.160
And we know what f of four is.

00:01:24.160 --> 00:01:26.750
F of four is equal to zero.

00:01:26.750 --> 00:01:31.050
So g of two is equal to f of
four, which is equal to zero.

00:01:31.050 --> 00:01:33.400
So notice, the corresponding
point has kind of

00:01:33.400 --> 00:01:36.610
gotten compressed in or
squeezed in or squished in,

00:01:36.610 --> 00:01:38.550
in the horizontal direction.

00:01:38.550 --> 00:01:39.860
And so what you see happening,

00:01:39.860 --> 00:01:41.660
at least on this side of the graph,

00:01:41.660 --> 00:01:43.960
is everything's happening
a little bit faster.

00:01:45.230 --> 00:01:46.610
Whatever was happening at a certain x,

00:01:46.610 --> 00:01:48.770
it's now happening at half of that x.

00:01:48.770 --> 00:01:51.590
So this side of the graph
is going to look something,

00:01:51.590 --> 00:01:54.060
try to draw it a little
bit better than that,

00:01:54.060 --> 00:01:56.300
it's going to look something

00:01:57.926 --> 00:01:59.503
like this,

00:01:59.503 --> 00:02:00.336
like this.

00:02:01.300 --> 00:02:03.850
Everything's happening twice as fast.

00:02:03.850 --> 00:02:06.180
And what happens when you go
in the negative direction?

00:02:06.180 --> 00:02:08.800
Well, think about what
g of negative two is.

00:02:08.800 --> 00:02:13.570
G of negative two is equal to
f of two times negative two,

00:02:13.570 --> 00:02:14.850
two times negative two,

00:02:14.850 --> 00:02:17.240
which is equal to f of negative four,

00:02:17.240 --> 00:02:18.960
which we see is also equal to zero.

00:02:18.960 --> 00:02:20.990
So g of negative two is zero.

00:02:20.990 --> 00:02:22.217
And you might be thinking,

00:02:22.217 --> 00:02:24.260
"Why did you pick two and negative two?"

00:02:24.260 --> 00:02:25.580
Well, the intuition is

00:02:25.580 --> 00:02:27.100
that things are going to be squeezed in.

00:02:27.100 --> 00:02:29.080
Things are happening twice as fast.

00:02:29.080 --> 00:02:30.730
So whatever was happening at x equals four

00:02:30.730 --> 00:02:32.020
is now going to happen at x equals two.

00:02:32.020 --> 00:02:33.390
Whatever is happening at
x equals negative four

00:02:33.390 --> 00:02:35.290
is now going to happen
at x equals negative two.

00:02:35.290 --> 00:02:37.620
And I saw that we were
at very clear points

00:02:37.620 --> 00:02:40.180
at x equals negative four
and x equals four on f,

00:02:40.180 --> 00:02:41.300
so I just took half of that

00:02:41.300 --> 00:02:43.970
to pick my x-values right over here.

00:02:43.970 --> 00:02:45.900
And then so what our graph is going

00:02:45.900 --> 00:02:48.203
to look like is something like this.

00:02:48.203 --> 00:02:49.920
It's going to look something like this.

00:02:49.920 --> 00:02:52.880
It's going to look like
it's been squished in

00:02:52.880 --> 00:02:55.980
from the right and the left.

00:02:55.980 --> 00:02:57.483
Now let's do another example.

00:02:59.100 --> 00:03:01.720
So now they've not only
given the graph of f,

00:03:01.720 --> 00:03:04.960
they've given an expression for it.

00:03:04.960 --> 00:03:09.190
What is the graph of g of x

00:03:09.190 --> 00:03:10.680
which is equal to this business?

00:03:10.680 --> 00:03:13.780
So pause this video, and
try to figure that out.

00:03:13.780 --> 00:03:15.440
All right, the key is to
figure out the relationship

00:03:15.440 --> 00:03:18.040
between f of x and g of x.

00:03:18.040 --> 00:03:21.830
And what we can see,
the main difference is,

00:03:21.830 --> 00:03:25.640
is instead of an x here in f
of x, we have an x over two.

00:03:25.640 --> 00:03:26.900
So everywhere there was an x,

00:03:26.900 --> 00:03:28.820
we've been replaced with an x over two.

00:03:28.820 --> 00:03:32.450
So another way of thinking
about it is g of x

00:03:32.450 --> 00:03:37.450
is equal to f of not
x but f of x over two.

00:03:38.260 --> 00:03:40.640
Or another way of
thinking about it, g of x

00:03:40.640 --> 00:03:42.423
is equal to f of 1/2x.

00:03:44.000 --> 00:03:47.280
And then we can do a
similar type of exercise.

00:03:47.280 --> 00:03:49.620
And they've given us
some interesting points,

00:03:49.620 --> 00:03:52.650
the points two, the point,

00:03:52.650 --> 00:03:55.230
or the point x equals two,
the point x equals four,

00:03:55.230 --> 00:03:57.310
and the point x equals six.

00:03:57.310 --> 00:03:58.930
So let's think about this.

00:03:58.930 --> 00:04:01.810
Last time, when it was g
of x is equal to two x,

00:04:01.810 --> 00:04:03.890
things were happening twice as fast.

00:04:03.890 --> 00:04:06.450
Now things are going
to happen half as fast.

00:04:06.450 --> 00:04:08.460
And so what I would do,

00:04:08.460 --> 00:04:11.090
let me just set up a little table here.

00:04:11.090 --> 00:04:13.760
The interesting x-values for me

00:04:13.760 --> 00:04:17.350
are the ones that if I take half of them,

00:04:17.350 --> 00:04:19.440
then I'm going to get one of these points.

00:04:19.440 --> 00:04:21.943
So actually let me write this, half, 1/2x,

00:04:23.330 --> 00:04:25.455
and then I can think about what g of x

00:04:25.455 --> 00:04:29.000
is equal to f of 1/2x is going to be.

00:04:29.000 --> 00:04:31.820
So I want my 1/2x to be,

00:04:31.820 --> 00:04:34.290
let's see, it could be two, four, and six,

00:04:34.290 --> 00:04:36.200
two, four, and six.

00:04:36.200 --> 00:04:37.970
And why did I pick those again?

00:04:37.970 --> 00:04:41.410
Well, it's very clear
what values f takes on

00:04:41.410 --> 00:04:43.050
at those points.

00:04:43.050 --> 00:04:48.050
And so if 1/2x is two,
then x is equal to four.

00:04:48.050 --> 00:04:50.860
If 1/2x is four, then x is equal to eight.

00:04:50.860 --> 00:04:54.020
If x is equal to 12, then 1/2x is six.

00:04:54.020 --> 00:04:56.730
And so then we could
say, all right, g of four

00:04:56.730 --> 00:04:59.760
is equal to f of two,

00:04:59.760 --> 00:05:01.810
which is equal to zero.

00:05:01.810 --> 00:05:03.580
That's why I picked two, four, and six.

00:05:03.580 --> 00:05:05.790
It's very easy to evaluate f of two,

00:05:05.790 --> 00:05:07.010
f of four, and f of six.

00:05:07.010 --> 00:05:09.190
They gave us those points very clearly.

00:05:09.190 --> 00:05:13.340
So g of eight is going to be equal to g,

00:05:13.340 --> 00:05:16.630
is going to be equal to f of
1/2 of eight, or f of four,

00:05:16.630 --> 00:05:19.130
which is equal to negative four.

00:05:19.130 --> 00:05:21.390
And then g of 12

00:05:21.390 --> 00:05:23.500
is equal to f of six,

00:05:23.500 --> 00:05:26.710
which is half of 12, which
is equal to zero again.

00:05:26.710 --> 00:05:28.110
So then we could plot these points,

00:05:28.110 --> 00:05:30.930
and we get a general sense
of the shape of the graph.

00:05:30.930 --> 00:05:32.300
So let's see,

00:05:32.300 --> 00:05:36.290
g of four is equal to zero,

00:05:36.290 --> 00:05:39.770
g of eight is equal to negative four

00:05:39.770 --> 00:05:41.750
right over there,

00:05:41.750 --> 00:05:44.510
and then g of 12 is equal to zero again.

00:05:44.510 --> 00:05:47.640
So everything has been stretched out.

00:05:47.640 --> 00:05:50.810
So there you go, it's been
stretched out in at least,

00:05:50.810 --> 00:05:54.100
in the horizontal direction
is one way to think about it,

00:05:54.100 --> 00:05:55.850
in the horizontal direction.

00:05:55.850 --> 00:05:58.890
And you can see that this point

00:05:58.890 --> 00:06:01.310
in f corresponds to this point in g.

00:06:01.310 --> 00:06:04.130
It's gotten twice as far from the origin

00:06:04.130 --> 00:06:06.640
because everything is
growing half as fast.

00:06:06.640 --> 00:06:08.360
You input an x, you take a half of it,

00:06:08.360 --> 00:06:10.160
and then you input it into f.

00:06:10.160 --> 00:06:13.000
And then this point right over here

00:06:13.000 --> 00:06:14.090
corresponds to this point.

00:06:14.090 --> 00:06:15.860
Instead of happening at
four, this vertex point,

00:06:15.860 --> 00:06:17.380
it's now happening at eight.

00:06:17.380 --> 00:06:19.630
And last but not least,

00:06:19.630 --> 00:06:22.500
this point right over here
corresponds to this point.

00:06:22.500 --> 00:06:25.440
Instead of happening at
six, it's happening at 12.

00:06:25.440 --> 00:06:27.630
Everything is getting stretched out.

00:06:27.630 --> 00:06:29.363
Let's do one more example.

00:06:30.730 --> 00:06:32.170
F of x is equal to all of this.

00:06:32.170 --> 00:06:34.150
We have to be careful,
there's a cube root over here.

00:06:34.150 --> 00:06:37.490
And g is a horizontally
scaled version of f.

00:06:37.490 --> 00:06:42.260
The functions are graphed where
f is solid and g is dashed.

00:06:42.260 --> 00:06:44.380
What is the equation of g?

00:06:44.380 --> 00:06:47.590
So pause this video, and see
if you can figure that out.

00:06:47.590 --> 00:06:48.920
All right, let's do this together,

00:06:48.920 --> 00:06:50.920
and it looks like they've
given us some points

00:06:50.920 --> 00:06:52.900
that seem to correspond with each other.

00:06:52.900 --> 00:06:56.020
To go from f to g, it looks
like these corresponding points

00:06:56.020 --> 00:06:59.810
have been squeezed in
closer to the origin.

00:06:59.810 --> 00:07:03.963
And what we can see is, is
that f of negative three,

00:07:04.930 --> 00:07:06.380
f of negative three

00:07:06.380 --> 00:07:10.253
seems to be equal to g of negative one.

00:07:11.370 --> 00:07:15.280
And f of six over here,

00:07:15.280 --> 00:07:19.540
f of six seems to be equal to g of two,

00:07:19.540 --> 00:07:21.410
g of two.

00:07:21.410 --> 00:07:22.850
Or another way to think about it,

00:07:22.850 --> 00:07:26.360
whatever x you input in g,
it looks like that's going

00:07:26.360 --> 00:07:29.980
to be equivalent to three times that x

00:07:29.980 --> 00:07:32.090
inputted into f.

00:07:32.090 --> 00:07:35.280
So g of x is equal to f of three x.

00:07:35.280 --> 00:07:37.770
And so if you want to
know the equation of g,

00:07:37.770 --> 00:07:40.300
we just evaluate f of three x.

00:07:40.300 --> 00:07:43.860
So f of three x is going to be equal to,

00:07:43.860 --> 00:07:47.360
and I could just actually
put an equal sign like this,

00:07:47.360 --> 00:07:50.410
f of three x is going to
be equal to negative three

00:07:50.410 --> 00:07:52.520
times the cube root of,

00:07:52.520 --> 00:07:54.930
instead of an x, I'll put
a three x right over there,

00:07:54.930 --> 00:07:58.780
three x plus two, and
then we have plus one.

00:07:58.780 --> 00:08:01.750
And that's it, that's
what g of x is equal to.

00:08:01.750 --> 00:08:03.530
It's equal to f of three x, which is that.

00:08:03.530 --> 00:08:06.190
We substituted this x with a three x.

00:08:06.190 --> 00:08:07.883
And we are done.


Scaling functions vertically: examples

WEBVTT
Kind: captions
Language: en

00:00:00.130 --> 00:00:02.000
- [Instructor] So, we're told that this

00:00:02.000 --> 00:00:04.600
is the graph of function
f right over here.

00:00:04.600 --> 00:00:08.260
And then they tell us that
function g is defined as

00:00:08.260 --> 00:00:11.340
g of x is equal to one third f of x.

00:00:11.340 --> 00:00:13.090
What is the graph of g?

00:00:13.090 --> 00:00:14.540
And if we were doing this on Khan Academy,

00:00:14.540 --> 00:00:16.720
this is a screenshot from our mobile app,

00:00:16.720 --> 00:00:17.940
it has multiple choices,

00:00:17.940 --> 00:00:19.680
but I thought we could
just try to sketch it.

00:00:19.680 --> 00:00:23.100
So pause this video, maybe in your mind,

00:00:23.100 --> 00:00:25.520
imagine what you think the graph
of g is going to look like,

00:00:25.520 --> 00:00:27.320
or at least how you would tackle it.

00:00:28.250 --> 00:00:31.670
All right, so g of x is
equal to one third f of x.

00:00:31.670 --> 00:00:36.300
So, for example, we can
see here that f of three

00:00:36.300 --> 00:00:38.520
is equal to negative three.

00:00:38.520 --> 00:00:41.870
So, g of three should be one third that,

00:00:41.870 --> 00:00:43.713
so it should be negative one.

00:00:44.780 --> 00:00:47.920
Likewise, so, g of three
would be right over there,

00:00:47.920 --> 00:00:51.980
and likewise, g of negative
three, what would that be?

00:00:51.980 --> 00:00:54.590
Well, f of negative three is three,

00:00:54.590 --> 00:00:57.820
so g of negative three is
going to be one third that,

00:00:57.820 --> 00:01:00.850
or it's going to be equal to one.

00:01:00.850 --> 00:01:03.630
F of zero is zero, one
third of that is still zero,

00:01:03.630 --> 00:01:06.230
so g of zero is still going
to be right over there.

00:01:06.230 --> 00:01:09.860
And we know that's going to
happen there and there as well,

00:01:09.860 --> 00:01:11.360
and so we already have a sense of what

00:01:11.360 --> 00:01:13.360
this graph is going to look like.

00:01:13.360 --> 00:01:16.930
The function g is going
to look something like,

00:01:16.930 --> 00:01:18.050
something like this.

00:01:18.050 --> 00:01:19.880
I'm just connecting the dots

00:01:19.880 --> 00:01:21.520
and they did give us some dots

00:01:21.520 --> 00:01:23.630
that we can use as reference points,

00:01:23.630 --> 00:01:27.020
so the graph of g is going
to look something like this.

00:01:27.020 --> 00:01:28.790
It gets a little bit flattened out

00:01:28.790 --> 00:01:32.730
or a little bit squooshed
or smooshed a little bit

00:01:32.730 --> 00:01:33.620
to look something like that

00:01:33.620 --> 00:01:35.590
and you would pick the
choice that looks like that.

00:01:35.590 --> 00:01:36.840
Let's do another example.

00:01:38.410 --> 00:01:42.897
So, here we are told
this the graph of f of x

00:01:42.897 --> 00:01:45.420
and it's defined by this expression.

00:01:45.420 --> 00:01:46.860
What is the graph of g of x,

00:01:46.860 --> 00:01:48.380
and g of x is this.

00:01:48.380 --> 00:01:50.630
So pause this video and
think about it again.

00:01:51.720 --> 00:01:53.930
All right, now the key realization is,

00:01:53.930 --> 00:01:56.700
is it looks like g of x is,

00:01:56.700 --> 00:01:58.900
if you were to take
all the terms of f of x

00:01:58.900 --> 00:02:01.340
and multiply it by two,

00:02:01.340 --> 00:02:02.210
or at least if you were to multiply

00:02:02.210 --> 00:02:03.970
the absolute value by two,

00:02:03.970 --> 00:02:06.560
and then if you were to multiply
this negative two by two.

00:02:06.560 --> 00:02:11.560
So it looks like g of x
is equal to two times,

00:02:11.680 --> 00:02:13.830
two times f of x.

00:02:13.830 --> 00:02:16.060
And we could even set
up a little table here,

00:02:16.060 --> 00:02:18.560
this is another of the way
that we can think about it.

00:02:18.560 --> 00:02:23.110
We can think about x, we
can think about f of x,

00:02:23.110 --> 00:02:24.700
and now we can think about g of x,

00:02:24.700 --> 00:02:26.530
which should be two times that.

00:02:26.530 --> 00:02:29.610
So we can see that when
x is equal to zero,

00:02:29.610 --> 00:02:31.920
f of x is equal to one,

00:02:31.920 --> 00:02:34.100
so g of x should be equal to two

00:02:34.100 --> 00:02:37.130
because it's two times f of x.

00:02:37.130 --> 00:02:39.190
So g of x is going to be equal to--

00:02:39.190 --> 00:02:42.450
Or g of zero, I should say,
is going to be equal to two.

00:02:42.450 --> 00:02:45.550
What about when at x equals,

00:02:45.550 --> 00:02:47.690
we'll say when x equals three.

00:02:47.690 --> 00:02:51.060
When x equals three, f
of x is negative two.

00:02:51.060 --> 00:02:54.040
G of x is going to be two times that,

00:02:54.040 --> 00:02:55.512
'cause it's two times f of x,

00:02:55.512 --> 00:02:57.150
so it's going to be negative four.

00:02:57.150 --> 00:02:59.820
So, g of x, or I should say g of three

00:02:59.820 --> 00:03:01.000
is going to be negative four.

00:03:01.000 --> 00:03:02.920
It's going to be right over there.

00:03:02.920 --> 00:03:05.560
And then maybe let's think
about one more point.

00:03:05.560 --> 00:03:10.560
So, f of five is equal to zero.

00:03:10.610 --> 00:03:12.330
G of five is going to be two times that,

00:03:12.330 --> 00:03:13.870
which is still going to be equal to zero,

00:03:13.870 --> 00:03:15.570
so it's going to be right over there.

00:03:15.570 --> 00:03:18.940
And so the graph is going
to look something like this,

00:03:18.940 --> 00:03:20.950
I'm just really just connecting,

00:03:20.950 --> 00:03:22.440
I'm just connecting the dots,

00:03:22.440 --> 00:03:24.380
trying to draw some straight lines.

00:03:24.380 --> 00:03:26.230
It's going to look something like this,

00:03:26.230 --> 00:03:28.170
you can see it's kind of stretched

00:03:28.170 --> 00:03:30.240
in the vertical direction.

00:03:30.240 --> 00:03:31.820
So, if you were doing
this on Khan Academy,

00:03:31.820 --> 00:03:32.653
it'd be multiple choice,

00:03:32.653 --> 00:03:34.890
you'd look for the graph
that looks like that.

00:03:34.890 --> 00:03:36.423
Let's do a few more examples.

00:03:37.710 --> 00:03:41.010
So, here we're given a function g

00:03:41.010 --> 00:03:44.150
is a vertically scaled version of f.

00:03:44.150 --> 00:03:47.910
So we can see that g is a
vertically scaled version of f.

00:03:47.910 --> 00:03:50.980
The functions are graphed
where f is a solid

00:03:50.980 --> 00:03:51.813
and g is dashed.

00:03:51.813 --> 00:03:52.840
Yeah, we see that.

00:03:52.840 --> 00:03:56.320
What is the equation of g in terms of f?

00:03:56.320 --> 00:03:58.670
So, pause this video and
try to think about it.

00:03:59.700 --> 00:04:01.900
Well, the way that I would
tackle this is once again,

00:04:01.900 --> 00:04:03.260
let's do it with a table

00:04:03.260 --> 00:04:06.150
and let's see the
relationship between f and g.

00:04:06.150 --> 00:04:08.930
So, this column is x,

00:04:08.930 --> 00:04:12.350
this column is f of x,

00:04:12.350 --> 00:04:15.685
and then this column is g of x.

00:04:15.685 --> 00:04:16.960
I'll make another column right over here.

00:04:16.960 --> 00:04:18.840
And so, let's see some interesting points.

00:04:18.840 --> 00:04:22.310
So, when, actually, I could pick zero,

00:04:22.310 --> 00:04:23.880
but zero is maybe less interesting

00:04:23.880 --> 00:04:25.270
than this point over here.

00:04:25.270 --> 00:04:28.600
So, this is when x is
equal to negative three.

00:04:28.600 --> 00:04:31.360
F of negative three is negative three.

00:04:31.360 --> 00:04:33.220
What is g of negative three?

00:04:33.220 --> 00:04:34.920
It looks like it is negative nine.

00:04:36.360 --> 00:04:39.900
When f is, when x is zero,

00:04:39.900 --> 00:04:42.520
f of zero is negative two.

00:04:42.520 --> 00:04:44.390
What is g of zero?

00:04:44.390 --> 00:04:47.820
It is equal to two negative six.

00:04:47.820 --> 00:04:50.550
And so we already see a pattern forming.

00:04:50.550 --> 00:04:53.840
Whatever f is, g is three times that.

00:04:53.840 --> 00:04:56.580
Whatever f is, g is three times that.

00:04:56.580 --> 00:04:59.460
And so we don't even actually
need these big columns,

00:04:59.460 --> 00:05:05.640
but we can see that g of x is
equal to three times f of x.

00:05:05.640 --> 00:05:08.201
So that is the equation
of g in terms of f.


Scaling functions introduction

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.570
- [Tutor] So this is a
screenshot of Desmos,

00:00:03.570 --> 00:00:05.200
it's an online graphing calculator,

00:00:05.200 --> 00:00:08.130
what we're gonna do is
use it to understand

00:00:08.130 --> 00:00:10.230
how we can go about scaling functions

00:00:10.230 --> 00:00:13.360
and I encourage you to go to
Desmos and try it on your own

00:00:13.360 --> 00:00:15.830
either during this video or after.

00:00:15.830 --> 00:00:18.290
So let's start with a
nice, interesting function,

00:00:18.290 --> 00:00:23.290
let's say f of x is equal
to the absolute value of x,

00:00:24.530 --> 00:00:26.024
so that's pretty straightforward.

00:00:26.024 --> 00:00:29.070
Now let's try to create a
scaled version of f of x,

00:00:29.070 --> 00:00:32.680
so we could say g of x is equal to,

00:00:32.680 --> 00:00:34.890
well, I'll start with
just absolute value of x,

00:00:34.890 --> 00:00:36.340
so it's the same as f of x,

00:00:36.340 --> 00:00:39.017
so we'll just trace the
g of x right on top of f,

00:00:39.017 --> 00:00:41.640
but now let's multiply it by sum constant,

00:00:41.640 --> 00:00:43.940
let's multiply it by two.

00:00:43.940 --> 00:00:46.324
So notice the difference
between g of x and f of x

00:00:46.324 --> 00:00:49.530
and you can see that g of
x is just two times f of x,

00:00:49.530 --> 00:00:51.080
in fact we can write it this way,

00:00:51.080 --> 00:00:54.505
we can write g of x is
equal to two times f of x,

00:00:54.505 --> 00:00:56.573
we get to the exact same place,

00:00:56.573 --> 00:01:01.573
but you can see that as our x increases,

00:01:02.160 --> 00:01:06.950
g of x increases twice as
fast, at least for positive xs

00:01:06.950 --> 00:01:10.300
on the right-hand side and
actually as x decreases,

00:01:10.300 --> 00:01:13.446
g of x also increases twice as fast,

00:01:13.446 --> 00:01:15.564
so is that just a coincidence
that we have a two here

00:01:15.564 --> 00:01:17.787
and it increased twice as fast?

00:01:17.787 --> 00:01:20.120
Well, let's put a three here,

00:01:20.120 --> 00:01:24.500
well now it looks like it's
increasing three times as fast

00:01:24.500 --> 00:01:26.758
and it does that in both directions.

00:01:26.758 --> 00:01:31.623
Now what if we were to
put a 0.5 here, 0.5?

00:01:31.623 --> 00:01:35.300
Well now it looks like it's
increasing half as fast

00:01:35.300 --> 00:01:39.020
and that makes sense, because
we are just multiplying,

00:01:39.020 --> 00:01:42.440
we are scaling how much our f of x is.

00:01:42.440 --> 00:01:46.610
So before when x equals
one, we got to one,

00:01:46.610 --> 00:01:49.410
but now when x equals one,
we only get to one half,

00:01:49.410 --> 00:01:51.690
before when x equals five, we got to five,

00:01:51.690 --> 00:01:55.790
now when we get to x equals
five, we only get to 2.5,

00:01:55.790 --> 00:01:57.245
so we're increasing half as fast,

00:01:57.245 --> 00:02:00.460
or we have half the slope.

00:02:00.460 --> 00:02:01.900
Now an interesting question

00:02:01.900 --> 00:02:03.910
to think about is what would happen

00:02:03.910 --> 00:02:07.900
if instead of it just being
an absolute value of x,

00:02:07.900 --> 00:02:12.010
let's say we were to have
a non-zero y intercept,

00:02:12.010 --> 00:02:16.950
so let's say, I don't know, plus six,

00:02:16.950 --> 00:02:21.780
so notice then when we change
this constant out front,

00:02:21.780 --> 00:02:23.740
it not only changes the slope,

00:02:23.740 --> 00:02:24.870
but it changes the y intercept,

00:02:24.870 --> 00:02:28.470
because we're multiplying
this entire expression by 0.5,

00:02:28.470 --> 00:02:29.800
so if you multiply it by one,

00:02:29.800 --> 00:02:31.615
we're back to where we got before

00:02:31.615 --> 00:02:33.426
and now if we multiply it by two,

00:02:33.426 --> 00:02:35.150
this should increase the y intercept,

00:02:35.150 --> 00:02:38.140
'cause remember we're
multiplying both of these terms

00:02:38.140 --> 00:02:42.250
by two and we see that, it
not only doubles the slope,

00:02:42.250 --> 00:02:44.720
but it also increases the y intercept.

00:02:44.720 --> 00:02:46.500
If we go to 0.5,

00:02:47.830 --> 00:02:51.000
not only did it decrease the
slope by a factor of one half,

00:02:51.000 --> 00:02:54.130
or I guess you could say
multiple the slope by one half,

00:02:54.130 --> 00:02:56.790
but it also made our y intercept

00:02:56.790 --> 00:02:59.580
be half of what it was before

00:02:59.580 --> 00:03:01.840
and we can see this more generally

00:03:01.840 --> 00:03:04.300
if we just put a general constant here

00:03:04.300 --> 00:03:06.640
and we can add a slider

00:03:06.640 --> 00:03:11.640
and actually let me make the
constant go from zero to 10

00:03:14.080 --> 00:03:18.124
with a step of, I don't know, 0.05,

00:03:18.124 --> 00:03:19.760
that's just how much does it increase

00:03:19.760 --> 00:03:21.320
every time you change the slider

00:03:21.320 --> 00:03:25.010
and notice when we increase our constant,

00:03:25.010 --> 00:03:26.420
not only we're getting narrower,

00:03:26.420 --> 00:03:29.720
'cause the magnitude of
the slope is being scaled,

00:03:29.720 --> 00:03:34.720
but our y intercept increases
and then as k decreases,

00:03:34.960 --> 00:03:37.930
our y intercept is being scaled down

00:03:37.930 --> 00:03:41.159
and our slope is being scaled down.

00:03:41.159 --> 00:03:43.590
Now that's one way that
we could go about scaling,

00:03:43.590 --> 00:03:45.490
but what if instead of multiplying

00:03:45.490 --> 00:03:47.730
our entire function by sum constant,

00:03:47.730 --> 00:03:52.600
we instead just replace the
x with a constant times x,

00:03:52.600 --> 00:03:54.625
so instead of k times f of x,

00:03:54.625 --> 00:03:58.892
what if we did it f of k times x?

00:03:58.892 --> 00:04:00.740
Another way to think about it is g of x

00:04:00.740 --> 00:04:03.395
is now equal to the absolute
value of kx plus six,

00:04:03.395 --> 00:04:05.240
what do you think is going to happen?

00:04:05.240 --> 00:04:07.380
Pause this video and think about it.

00:04:07.380 --> 00:04:10.450
Well now when we increase k,

00:04:10.450 --> 00:04:13.133
notice it has no impact
on our y intercept,

00:04:13.133 --> 00:04:15.557
because it's not scaling the y intercept,

00:04:15.557 --> 00:04:17.780
but it does have an impact on slope,

00:04:17.780 --> 00:04:20.750
when k goes from one to two,

00:04:20.750 --> 00:04:24.455
once again we are now
increasing twice as fast

00:04:24.455 --> 00:04:28.265
and then when k goes from one to one half,

00:04:28.265 --> 00:04:31.490
we're now increasing half as fast.

00:04:31.490 --> 00:04:33.236
Now this is with an
absolute value function,

00:04:33.236 --> 00:04:35.340
what if we did it with a
different type of function,

00:04:35.340 --> 00:04:37.500
let's say we did it with a quadratic?

00:04:37.500 --> 00:04:41.180
So two minus x squared,

00:04:41.180 --> 00:04:43.880
let me scroll down a little bit

00:04:43.880 --> 00:04:48.170
and so you can see when k
equals one, these are the same

00:04:48.170 --> 00:04:50.214
and now if we increase our k,

00:04:50.214 --> 00:04:52.913
let's say we increase our k to two,

00:04:53.810 --> 00:04:58.690
notice our parabola is
in this case decreasing

00:04:58.690 --> 00:05:00.690
as we get further and further from zero

00:05:00.690 --> 00:05:02.066
at a faster and faster rate,

00:05:02.066 --> 00:05:05.502
that's because what you would
have seen at x equals two,

00:05:05.502 --> 00:05:07.620
you're now seeing at x equals one,

00:05:07.620 --> 00:05:11.210
because you are multiplying two times that

00:05:11.210 --> 00:05:15.130
and so then if we go between zero and one,

00:05:15.130 --> 00:05:18.090
notice on either side of zero,

00:05:18.090 --> 00:05:21.396
our parabola is decreasing
at a lower rate,

00:05:21.396 --> 00:05:24.384
it's a changing rate, but
it's a lower changing rate,

00:05:24.384 --> 00:05:26.780
I guess you could put it that way

00:05:26.780 --> 00:05:28.560
and we could also try just to see

00:05:28.560 --> 00:05:30.640
what happens with our parabola here,

00:05:30.640 --> 00:05:34.778
if instead of doing kx, we
once again put the k out front,

00:05:34.778 --> 00:05:36.400
what is that going to do?

00:05:36.400 --> 00:05:39.160
And notice that is
changing not only how fast

00:05:39.160 --> 00:05:41.173
the curve changes at different points,

00:05:41.173 --> 00:05:43.246
but it's now also
changing the y intercept,

00:05:43.246 --> 00:05:46.290
because we are now
scaling that y intercept.

00:05:46.290 --> 00:05:47.280
So I'll leave you there,

00:05:47.280 --> 00:05:49.434
this is just the beginning
of thinking about scaling,

00:05:49.434 --> 00:05:51.970
I really want you to
build an intuitive sense

00:05:51.970 --> 00:05:52.827
of what is going on here

00:05:52.827 --> 00:05:55.940
and really think about
mathematically why it makes sense

00:05:55.940 --> 00:05:58.600
and go on to Desmos and
play around with it yourself

00:05:58.600 --> 00:06:02.153
and also try other types of
functions and see what happens.


Even and odd functions: Find the mistake

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:03.060
- [Instructor] We are told
Jayden was asked to determine

00:00:03.060 --> 00:00:07.720
whether f of x is equal to
x minus the cube root of x

00:00:07.720 --> 00:00:10.810
is even, odd, or neither.

00:00:10.810 --> 00:00:12.880
Here is his work.

00:00:12.880 --> 00:00:15.210
Is Jayden's work correct?

00:00:15.210 --> 00:00:19.860
If not, what is the first step
where Jayden made a mistake?

00:00:19.860 --> 00:00:22.210
So pause this video and
review Jayden's work,

00:00:22.210 --> 00:00:23.580
and see if it's correct,

00:00:23.580 --> 00:00:27.380
or if it's not correct tell
me where it's not correct.

00:00:27.380 --> 00:00:29.700
All right, now let's work this together.

00:00:29.700 --> 00:00:31.470
So, let's see, just to remind ourselves

00:00:31.470 --> 00:00:33.670
what Jayden's trying to
do, he's trying to decide,

00:00:33.670 --> 00:00:37.760
whether f of x is even, odd, or neither.

00:00:37.760 --> 00:00:40.530
And f of x is expressed, or is defined,

00:00:40.530 --> 00:00:42.830
as x minus the cube root of x.

00:00:42.830 --> 00:00:45.230
So let's see, the first
thing that Jayden did is

00:00:45.230 --> 00:00:48.820
he's trying to figure out
what is f of negative x?

00:00:48.820 --> 00:00:51.340
Because remember, if f of negative x

00:00:51.340 --> 00:00:54.586
is equal to f of x, we are even,

00:00:54.586 --> 00:00:56.793
and if f of negative x

00:00:56.793 --> 00:01:01.793
is equal to negative f
of x, then we are odd.

00:01:01.920 --> 00:01:02.970
So it makes sense for him

00:01:02.970 --> 00:01:05.580
to find the expression
for f of negative x.

00:01:05.580 --> 00:01:08.630
So he tries to evaluate f of negative x,

00:01:08.630 --> 00:01:10.250
and when he does that,

00:01:10.250 --> 00:01:12.700
everywhere where he sees an x in f of x,

00:01:12.700 --> 00:01:14.760
he replaces it with a negative x.

00:01:14.760 --> 00:01:16.510
So that seems good.

00:01:16.510 --> 00:01:19.530
And then, let's see, this
becomes a negative x,

00:01:19.530 --> 00:01:21.000
that makes sense, minus,

00:01:21.000 --> 00:01:24.160
and then, a negative x under the radical,

00:01:24.160 --> 00:01:25.780
and this is a cube root right over here,

00:01:25.780 --> 00:01:28.710
that's the same thing
as negative one times x.

00:01:28.710 --> 00:01:30.660
The cube root of negative
one is negative one.

00:01:30.660 --> 00:01:33.390
So he takes that negative
out of the radical,

00:01:33.390 --> 00:01:34.560
out of the cube root.

00:01:34.560 --> 00:01:38.560
So this makes sense, and
so then he has a negative x

00:01:38.560 --> 00:01:41.110
and you subtract a negative,
you get a positive.

00:01:41.110 --> 00:01:43.460
So then that makes sense.

00:01:43.460 --> 00:01:46.510
And then, the next thing he
says is, or he's trying to do,

00:01:46.510 --> 00:01:48.400
is check if f of negative x

00:01:48.400 --> 00:01:51.360
is equal to f of x or f of negative x.

00:01:51.360 --> 00:01:53.800
So he's gonna check whether
this is equal to one of them.

00:01:53.800 --> 00:01:58.440
And so here Jayden says, negative
x plus the cube root of x,

00:01:58.440 --> 00:02:02.070
so that's what f of negative
x, what he evaluated it to be,

00:02:02.070 --> 00:02:06.750
isn't the same as f of x, now
let's see is that the case?

00:02:06.750 --> 00:02:09.644
Is it not the same as f of x?

00:02:09.644 --> 00:02:12.691
Yup it's definitely, it's
not the same as f of x,

00:02:12.691 --> 00:02:16.640
or negative f of x which is equal to

00:02:16.640 --> 00:02:20.288
negative x minus the cube root of x.

00:02:20.288 --> 00:02:22.300
Now that seems a little bit fishy.

00:02:22.300 --> 00:02:25.920
Did he do the right
thing, right over here?

00:02:25.920 --> 00:02:27.640
Is negative f of x equal to

00:02:27.640 --> 00:02:29.950
negative x minus the cube root of x?

00:02:29.950 --> 00:02:32.770
Let's see, negative of f of x

00:02:32.770 --> 00:02:35.380
is going to be a negative
times this entire expression,

00:02:35.380 --> 00:02:37.270
it's going to be a negative up front,

00:02:37.270 --> 00:02:41.400
times x minus the cube root of x,

00:02:41.400 --> 00:02:42.890
and so this is going to be equal to,

00:02:42.890 --> 00:02:45.610
you distribute the negative sign,

00:02:45.610 --> 00:02:50.290
you get negative x plus
the cube root of x.

00:02:50.290 --> 00:02:53.365
So Jayden calculated the wrong

00:02:53.365 --> 00:02:56.730
negative f of x right over here.

00:02:56.730 --> 00:03:01.730
So, he isn't right that negative
x plus the cube root of x,

00:03:01.845 --> 00:03:04.848
it is actually the same
as negative f of x.

00:03:04.848 --> 00:03:07.620
So he's wrong right over here.

00:03:07.620 --> 00:03:12.200
So Jayden's mistake is right
over here, really it looks like

00:03:12.200 --> 00:03:15.460
he didn't evaluate
negative f of x correctly.

00:03:15.460 --> 00:03:17.560
So Jayden's work, is
Jayden's work correct?

00:03:17.560 --> 00:03:18.540
No.

00:03:18.540 --> 00:03:20.970
If not, what is the first step
where Jayden made a mistake?

00:03:20.970 --> 00:03:23.130
Well it would be step two.

00:03:23.130 --> 00:03:24.686
What he should have said is,

00:03:24.686 --> 00:03:28.010
it actually is the same
as negative f of x,

00:03:28.010 --> 00:03:30.177
and so therefore his conclusion should be

00:03:30.177 --> 00:03:33.273
that f of x is odd.


Even and odd functions: Equations

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.043
- [Instructor] We are asked

00:00:01.043 --> 00:00:04.900
are the following functions
even, odd, or neither?

00:00:04.900 --> 00:00:07.240
So pause this video and try
to work that out on your own

00:00:07.240 --> 00:00:09.720
before we work through it together.

00:00:09.720 --> 00:00:12.270
All right, now let's just remind ourselves

00:00:12.270 --> 00:00:15.010
of a definition for
even and odd functions.

00:00:15.010 --> 00:00:16.720
One way to think about it is what happens

00:00:16.720 --> 00:00:19.000
when you take f of negative x?

00:00:19.000 --> 00:00:23.460
If f of negative x is equal
to the function again,

00:00:23.460 --> 00:00:26.190
then we're dealing with an even function.

00:00:26.190 --> 00:00:29.160
If we evaluate f of negative x,

00:00:29.160 --> 00:00:30.210
instead of getting the function,

00:00:30.210 --> 00:00:32.640
we get the negative of the function,

00:00:32.640 --> 00:00:34.590
then we're dealing with an odd function.

00:00:34.590 --> 00:00:37.610
And if neither of these
are true it is neither.

00:00:37.610 --> 00:00:40.400
So let's go to this first
one right over here,

00:00:40.400 --> 00:00:45.400
f of x is equal to five over
three minus x to the fourth,

00:00:45.410 --> 00:00:47.480
and the best way I can
think about tackling this

00:00:47.480 --> 00:00:48.380
is let's just evaluate

00:00:48.380 --> 00:00:51.130
what f of negative x would be equal to.

00:00:51.130 --> 00:00:55.760
That would be equal to
five over three minus

00:00:55.760 --> 00:00:56.970
and everywhere we see an x,

00:00:56.970 --> 00:00:59.220
we're gonna replace
that with a negative x,

00:00:59.220 --> 00:01:00.870
to the fourth power.

00:01:00.870 --> 00:01:03.370
Now what is negative
x to the fourth power?

00:01:03.370 --> 00:01:06.480
Well if you multiply a
negative times a negative

00:01:06.480 --> 00:01:08.970
times a negative, how
many times did I do that?

00:01:08.970 --> 00:01:10.910
If you take a negative
to the fourth power,

00:01:10.910 --> 00:01:12.280
you're going to get a positive,

00:01:12.280 --> 00:01:16.670
so that's going to be
equal to five over three

00:01:16.670 --> 00:01:21.617
minus x to the fourth, which
is once again equal to f of x

00:01:22.850 --> 00:01:24.690
and so this first one right over here,

00:01:24.690 --> 00:01:28.880
f of negative x is equal to
f of x, it is clearly even.

00:01:28.880 --> 00:01:31.380
Let's do another example.

00:01:31.380 --> 00:01:34.960
So this one right over here, g of x,

00:01:34.960 --> 00:01:36.710
let's just evaluate g of negative x

00:01:36.710 --> 00:01:38.290
and at any point, you feel inspired

00:01:38.290 --> 00:01:40.030
and you didn't figure
it out the first time,

00:01:40.030 --> 00:01:42.410
pause the video again and try
to work it out on your own.

00:01:42.410 --> 00:01:47.100
Well g of negative x is
equal to one over negative x

00:01:47.100 --> 00:01:51.637
plus the cube root of negative x

00:01:53.580 --> 00:01:57.470
and let's see, can we simplify this any?

00:01:57.470 --> 00:02:01.760
Well we could rewrite this
as the negative of one over x

00:02:01.760 --> 00:02:04.640
and then yeah, I could view negative x

00:02:04.640 --> 00:02:09.330
as the same thing as negative one times x

00:02:09.330 --> 00:02:10.970
and so we can factor out,

00:02:10.970 --> 00:02:12.850
or I should say we could
take the negative one

00:02:12.850 --> 00:02:14.040
out of the radical.

00:02:14.040 --> 00:02:15.960
What is the cube root of negative one?

00:02:15.960 --> 00:02:17.050
Well it's negative one,

00:02:17.050 --> 00:02:20.410
so we could say minus
one times the cube root

00:02:20.410 --> 00:02:25.410
or we could just say the
negative of the cube root of x

00:02:25.720 --> 00:02:28.930
and then we can factor out a negative,

00:02:28.930 --> 00:02:32.770
so this is going to be equal
to negative of one over x

00:02:32.770 --> 00:02:35.870
plus the cube root of x,

00:02:35.870 --> 00:02:38.370
which is equal to the negative of g of x,

00:02:38.370 --> 00:02:40.920
which is equal to the negative of g of x.

00:02:40.920 --> 00:02:44.010
And so this is odd,

00:02:44.010 --> 00:02:47.130
f of negative x is equal
to the negative of f of x,

00:02:47.130 --> 00:02:48.320
or in this case it's g of x,

00:02:48.320 --> 00:02:51.400
g of negative x is equal
to the negative of g of x.

00:02:51.400 --> 00:02:53.280
Let's do the third one.

00:02:53.280 --> 00:02:55.930
So here we've got h of x

00:02:56.800 --> 00:02:59.390
and let's just evaluate h of negative x.

00:02:59.390 --> 00:03:03.730
h of negative x is equal
to two to the negative x

00:03:03.730 --> 00:03:07.350
plus two to the negative of negative x,

00:03:07.350 --> 00:03:09.810
which would be two to the positive x.

00:03:09.810 --> 00:03:12.370
Well this is the same thing
as our original h of x.

00:03:12.370 --> 00:03:14.410
This is just equal to h of x.

00:03:14.410 --> 00:03:16.420
You just swap these two terms

00:03:16.420 --> 00:03:19.480
and so this is clearly even.

00:03:19.480 --> 00:03:23.900
And then last but not
least, we have j of x,

00:03:23.900 --> 00:03:25.383
so let's evaluate j of,

00:03:25.383 --> 00:03:27.190
why did I write y?

00:03:27.190 --> 00:03:31.150
Let's evaluate j of negative x

00:03:31.150 --> 00:03:36.150
is equal to negative x
over one minus negative x,

00:03:36.770 --> 00:03:40.810
which is equal to negative
x over one plus x,

00:03:40.810 --> 00:03:44.230
and let's see, there's no clear way

00:03:44.230 --> 00:03:45.720
of factoring out a negative

00:03:45.720 --> 00:03:46.970
or doing something interesting

00:03:46.970 --> 00:03:48.870
where I get either back to j of x,

00:03:48.870 --> 00:03:50.980
or I get to negative j of x,

00:03:50.980 --> 00:03:54.000
so this one is neither

00:03:54.000 --> 00:03:54.833
and we're done.


Even and odd functions: Tables

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:03.770
- [Instructor] We're told
this table defines function f.

00:00:03.770 --> 00:00:04.780
All right.

00:00:04.780 --> 00:00:08.680
For every x, they give us
the corresponding f of x.

00:00:08.680 --> 00:00:13.460
According to the table, is
f even, odd, or neither?

00:00:13.460 --> 00:00:14.650
So pause this video and see

00:00:14.650 --> 00:00:17.090
if you can figure that out on your own.

00:00:17.090 --> 00:00:18.960
All right, now let's
work on this together.

00:00:18.960 --> 00:00:20.930
So let's just remind
ourselves the definition

00:00:20.930 --> 00:00:22.690
of even and odd.

00:00:22.690 --> 00:00:26.490
One definition that we can
think of is that f of x,

00:00:26.490 --> 00:00:30.620
if f of x is equal to f of negative x,

00:00:30.620 --> 00:00:34.410
then we're dealing with an even function.

00:00:34.410 --> 00:00:37.460
And if f of x is equal

00:00:37.460 --> 00:00:40.690
to the negative of f of negative x,

00:00:40.690 --> 00:00:44.210
or another way of saying
that, if f of negative x.

00:00:44.210 --> 00:00:46.830
If f of negative x, instead
of it being equal to f of x,

00:00:46.830 --> 00:00:48.870
it's equal to negative f of x.

00:00:48.870 --> 00:00:50.850
These last two are equivalent.

00:00:50.850 --> 00:00:52.170
Then in these situations,

00:00:52.170 --> 00:00:53.840
we are dealing with an odd function.

00:00:53.840 --> 00:00:56.130
And if neither of these are true,

00:00:56.130 --> 00:00:58.010
then we're dealing with neither.

00:00:58.010 --> 00:01:00.290
So what about what's going on over here?

00:01:00.290 --> 00:01:01.800
So let's see.

00:01:01.800 --> 00:01:05.680
F of negative seven is
equal to negative one.

00:01:05.680 --> 00:01:08.470
What about f of the
negative of negative seven?

00:01:08.470 --> 00:01:10.290
Well, that would be f of seven.

00:01:10.290 --> 00:01:14.030
And we see f of seven here is
also equal to negative one.

00:01:14.030 --> 00:01:17.280
So at least in that case and that case,

00:01:17.280 --> 00:01:19.670
if we think of x as seven,

00:01:19.670 --> 00:01:24.590
f of x is equal to f of negative x.

00:01:24.590 --> 00:01:25.890
So it works for that.

00:01:25.890 --> 00:01:28.750
It also works for
negative three and three.

00:01:28.750 --> 00:01:31.330
F of three is equal to
f of negative three.

00:01:31.330 --> 00:01:33.030
They're both equal to two.

00:01:33.030 --> 00:01:35.900
And you can see and you can
kind of visualize in your head

00:01:35.900 --> 00:01:39.500
that we have symmetry around the y-axis.

00:01:39.500 --> 00:01:42.960
And so this looks like an even function.

00:01:42.960 --> 00:01:46.010
So I will circle that in.

00:01:46.010 --> 00:01:47.260
Let's do another example.

00:01:48.410 --> 00:01:51.020
So here, once again, the
table defines function f.

00:01:51.020 --> 00:01:52.560
It's a different function f.

00:01:52.560 --> 00:01:54.840
Is this function even, odd, or neither?

00:01:54.840 --> 00:01:57.490
So pause this video and
try to think about it.

00:01:57.490 --> 00:01:59.980
All right, so let's
just try a few examples.

00:01:59.980 --> 00:02:03.820
So here we have f of five is equal to two.

00:02:03.820 --> 00:02:06.100
F of five is equal to two.

00:02:06.100 --> 00:02:08.650
What is f of negative five?

00:02:08.650 --> 00:02:10.820
F of negative five.

00:02:10.820 --> 00:02:12.610
Not only is it not equal to two,

00:02:12.610 --> 00:02:13.920
it would have to be equal to two

00:02:13.920 --> 00:02:15.540
if this was an even function.

00:02:15.540 --> 00:02:17.800
And it would be equal to negative two

00:02:17.800 --> 00:02:19.860
if this was an odd
function, but it's neither.

00:02:19.860 --> 00:02:22.950
So we very clearly see just
looking at that data point

00:02:22.950 --> 00:02:26.250
that this can neither be even, nor odd.

00:02:26.250 --> 00:02:30.040
So I would say neither or
neither right over here.

00:02:30.040 --> 00:02:31.340
Let's do one more example.

00:02:32.400 --> 00:02:34.540
Once again, the table defines function f.

00:02:34.540 --> 00:02:36.690
According to the table, is
it even, odd, or neither?

00:02:36.690 --> 00:02:37.630
Pause the video again.

00:02:37.630 --> 00:02:38.533
Try to answer it.

00:02:39.810 --> 00:02:42.860
All right, so actually
let's just start over here.

00:02:42.860 --> 00:02:47.070
So we have f of four is
equal to negative eight.

00:02:47.070 --> 00:02:49.760
What is f of negative four?

00:02:49.760 --> 00:02:51.200
And the whole idea here is I wanna say,

00:02:51.200 --> 00:02:53.050
okay, if f of x is equal to something,

00:02:53.050 --> 00:02:54.770
what is f of negative x?

00:02:54.770 --> 00:02:56.920
Well, they luckily give
us f of negative four.

00:02:56.920 --> 00:02:59.330
It is equal to eight.

00:02:59.330 --> 00:03:01.810
So it looks like it's not equal to f of x.

00:03:01.810 --> 00:03:03.730
It's equal to the negative of f of x.

00:03:03.730 --> 00:03:08.010
This is equal to the
negative of f of four.

00:03:08.010 --> 00:03:09.980
So on that data point alone,

00:03:09.980 --> 00:03:13.020
at least that data point
satisfies it being odd.

00:03:13.020 --> 00:03:15.490
It's equal to the negative of f of x.

00:03:15.490 --> 00:03:18.400
But now let's try the other
points just to make sure.

00:03:18.400 --> 00:03:22.020
So f of one is equal to five.

00:03:22.020 --> 00:03:24.033
What is f of negative one?

00:03:24.900 --> 00:03:27.350
Well, it is equal to negative five.

00:03:27.350 --> 00:03:29.780
Once again, f of negative x is equal

00:03:29.780 --> 00:03:32.240
to the negative of f of x.

00:03:32.240 --> 00:03:33.370
So that checks out.

00:03:33.370 --> 00:03:35.940
And then f of zero,

00:03:35.940 --> 00:03:38.870
well, f of zero is of
course equal to zero.

00:03:38.870 --> 00:03:40.840
But of course if you say

00:03:40.840 --> 00:03:44.930
what is the negative of f of,

00:03:44.930 --> 00:03:46.940
if you say what f of negative of zero,

00:03:46.940 --> 00:03:48.520
well, that's still f of zero.

00:03:48.520 --> 00:03:51.190
And then if you were to
take the negative of zero,

00:03:51.190 --> 00:03:52.730
that's still zero.

00:03:52.730 --> 00:03:54.160
So you could view this.

00:03:54.160 --> 00:03:56.610
This is consistent still with being odd.

00:03:56.610 --> 00:04:00.990
This you could view as the
negative of f of negative zero,

00:04:00.990 --> 00:04:03.120
which of course is still going to be zero.

00:04:03.120 --> 00:04:08.023
So this one is looking
pretty good that it is odd.