Powers of products & quotients (integer exponents)

WEBVTT
Kind: captions
Language: en

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- [Instructor] Do some example,

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raising exponents or products of exponents

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to various powers,

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especially when we're dealing
with integer exponents.

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So let's say we have three
to the negative eight

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times seven to the third,

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and we wanna raise that
to the negative two power,

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and I want you to pause this video

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and see if you could
simplify this on your own.

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So the key realization here,

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there's couple of ways
that you can tackle it,

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but the key thing to realize

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is if you have the product of two things,

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and then you're raising that
to some type of a exponent,

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that is going to be the same thing

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as raising each of these
things to that exponent,

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and then taking the product.

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So this is going to be the same thing

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as three to the negative eight,

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and then that to the negative two

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times seven to the third
to the negative two,

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so I'll do seven to the
third right over here.

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And if I wanna simplify this,

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three to the negative
eight to the negative two,

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we have the other exponent property

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that if you're raising to an exponent

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and then raising that whole
thing to another exponent,

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then you can just multiply the exponents.

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So this is going to be
three to the negative eight

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times negative two power.

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Well, negative eight times negative two

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is positive 16,

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so this is gonna be
three to the 16th power

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right over there,

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and then this part right over here,

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seven to the third to the negative two.

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that's gonna be seven to the three

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times negative two,

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which is seven to the
negative sixth power.

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So that is seven to the negative six,

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and this would be about as
much as you could simplify.

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You could rewrite it different ways.

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Seven to the negative
six is the same thing

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as one over seven to the sixth,

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so you could write it
like three to the 16th.

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We'll use that same shade of blue,

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three to the 16th

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over seven to the sixth,

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but these two are equivalent,

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and there's other ways that
you could have tackled this.

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You could have said

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that this original thing right over here,

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this is the same thing as,

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three to the negative eight

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is the same thing as one
over three to the eighth,

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so you could have said
this is the same thing

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as seven to the third

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over three to the eighth,

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and then you're raising
that to the negative two,

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in which case you'd raise this
numerator to the negative two

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and the denominator to negative two,

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but you would have gotten
to the exact same place.

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Let's do another one of these.

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So let's say

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that we have got

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A to the negative two times
eight to the seventh power,

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and we wanna raise all of
that to the second power.

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Well, like before, I can
raise each of these things

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to the second power,

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so this is the same thing
as A to the negative two

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to the second power

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times this thing to the second power.

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Eight to the seventh to the second power,

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and then here, negative two
times two is negative four,

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so that's A to the negative four times,

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eight to the seven times two is 14,

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eight to the 14th power.

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In other videos, we go into more depth

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about why this should
hopefully make intuitive sense.

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Here you have eight to the seventh

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times eight to the seventh.

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Well, you would then
add the two exponents,

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and you would get to eight to the 14th,

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so however many times you
have eight to the seventh,

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you would just keep adding the exponents,

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or you would multiply by
seven that many times.

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Hopefully that didn't sound too confusing,

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but the general idea is if you
raise something to exponent

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and then another exponent,

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you can multiply those exponents.

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Let's do one more example

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where we are dealing with quotients,

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which that first example

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could have even been perceived as.

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So let's say we have

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two to the negative 10

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divided by four squared,

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and we're gonna raise all of
that to the seventh power.

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Well, this is equivalent
to two to the negative 10

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raised to the seventh power

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over four squared raised
to the seventh power,

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so if you have the
difference of two things

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and you're raising it to some power,

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that's the same thing as a numerator

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raised to that power

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divided by the denominator
raised to that power.

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Well, what's our numerator going to be?

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Well, we've done this drill before.

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It'd be two to the negative
10 times seventh power,

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so this would be equal to two
to the negative 70th power,

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and then in the denominator,

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four to the second power,

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then that raised to the seventh power.

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Well, two times seven is 14,

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so that's going to be four

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to the 17th power.

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Now, we actually could think

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about simplifying this even more.

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There's multiple ways that
you could rewrite this,

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but one thing you could
do is say, "Hey, look,

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"four is a power of two."

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So you could rewrite this as

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this is equal to two to the
negative 70th power over,

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instead of writing four to the 17th power,

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why did I write the 17th power?

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It should be four to the 14th power.

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Let me correct that.

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Instead of writing four to the 14th power,

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I instead could write,

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so this is two, get the colors right.

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This is two to the negative 70th over,

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instead of writing four,

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I could write two squared
to the 14th power.

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Four is the same thing as two squared,

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and so now I can rewrite this whole thing

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as two to the negative 70th power over,

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well, two to the second,

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and then that to the 14th,

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well, that's two to the 28th power,

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two to the 28th power.

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And so can I simplify this even more?

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Well, this is going to be equal

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to two to the,

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if I'm taking a quotient
with the same base,

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I can subtract the exponent.

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So it's gonna be negative 70.

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It's going to be negative 70

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minus 28th power, minus 28,

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and so this is going to simply

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two to the negative 98th power,

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and that's another way of
viewing the same expression.


Multiplying & dividing powers (integer exponents)

WEBVTT
Kind: captions
Language: en

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- [Narrator] Let's get some practice

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with our exponent properties,

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especially when we have integer exponents.

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So, let's think about what
four to the negative three

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times four to the fifth power

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is going to be equal to.

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And I encourage you to pause the video

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and think about it on your own.

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Well there's a couple of ways to do this.

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See look, I'm multiplying two things

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that have the same base,

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so this is going to be that base, four.

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And then I add the exponents.

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Four to the negative three plus five power

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which is equal to four
to the second power.

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And that's just a straight
forward exponent property,

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but you can also think about

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why does that actually make sense.

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Four to the negative 3 power,

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that is one over four to the third power,

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or you could view that as

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one over four times four times four.

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And then four to the fifth,

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that's five fours being
multiplied together.

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So it's times four times four

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times four times four times four.

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And so notice, when you multiply this out,

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you're going to have five
fours in the numerator

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and three fours in the denominator.

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And so, three of these in the denominator

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are going to cancel out

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with three of these in the numerator.

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And so you're going to be left

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with five minus three,

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or negative three plus five fours.

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So this four times four

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is the same thing as four squared.

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Now let's do one with variables.

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So let's say that you have

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A to the negative fourth power

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times A to the,

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let's say, A squared.

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What is that going to be?

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Well once again, you have the same base,

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in this case it's A, and so
since I'm multiplying them,

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you can just add the exponents.

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So it's going to be A to the
negative four plus two power.

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Which is equal to A to
the negative two power.

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And once again, it should make sense.

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This right over here,

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that is one over A times A times A times A

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and then this is

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times A times A,

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so that cancels with that,
that cancels with that,

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and you're still left
with one over A times A,

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which is the same thing as
A to the negative two power.

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Now, let's do it with some quotients.

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So, what if I were to ask you,

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what is 12 to the negative seven

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divided by 12 to the negative five power?

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Well, when you're dividing,

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you subtract exponents if
you have the same base.

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So, this is going to be equal

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to 12 to the negative seven

00:02:49.288 --> 00:02:52.004
minus negative five power.

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You're subtracting the bottom exponent

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and so, this is going to
be equal to 12 to the,

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subtracting a negative is the same thing

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as adding the positive,

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twelve to the negative two power.

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And once again, we just
have to think about,

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why does this actually make sense?

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Well, you could actually rewrite this.

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12 to the negative seven

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divided by 12 to the negative five,

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that's the same thing as
12 to the negative seven

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times 12 to the fifth power.

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If we take the reciprocal
of this right over here,

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you would make exponent positive

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and then you would get
exactly what we were doing

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in those previous examples with products.

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And so, let's just do one more

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with variables for good measure.

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Let's say I have X to the
negative twentieth power

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divided by X to the fifth power.

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Well once again, we have the same base

00:03:49.916 --> 00:03:51.379
and we're taking a quotient.

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So, this is going to be X to
the negative 20 minus five

00:03:55.645 --> 00:03:59.075
cause we have this one right
over here in the denominator.

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So, this is going to be equal

00:04:00.166 --> 00:04:03.792
to X to the negative twenty-fifth power.

00:04:03.792 --> 00:04:05.835
And once again, you could
view our original expression

00:04:05.835 --> 00:04:08.331
as X to the negative twentieth

00:04:08.331 --> 00:04:10.793
and having an X to the
fifth in the denominator

00:04:10.793 --> 00:04:12.482
dividing by X to the fifth

00:04:12.482 --> 00:04:14.241
is the same thing as multiplying

00:04:14.241 --> 00:04:16.095
by X to the negative five.

00:04:16.095 --> 00:04:18.384
So here you just add the exponents

00:04:18.384 --> 00:04:19.584
and once again you would get

00:04:19.584 --> 00:04:22.667
X to the negative twenty-fifth power.


Graphing a shifted and stretched absolute value function

WEBVTT
Kind: captions
Language: en

00:00:00.238 --> 00:00:02.720
- [Instructor] So we're asked
to graph f of x is equal to

00:00:02.720 --> 00:00:07.422
two times the absolute value
of x plus three, plus two.

00:00:07.422 --> 00:00:09.088
And what they've already graphed for us,

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this right over here,

00:00:10.278 --> 00:00:14.445
this is the graph of y is equal
to the absolute value of x.

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So let's do this through a
series of transformations.

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So the next thing I wanna graph,

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let's see if we can graph y.

00:00:24.047 --> 00:00:28.838
Y is equal is to the absolute
value of x plus three.

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Now in previous videos
we have talked about it.

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If you replace your x,

00:00:32.956 --> 00:00:35.454
with an x plus three,

00:00:35.454 --> 00:00:40.137
this is going to shift your
graph to the left by three.

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You could view this as the same thing as

00:00:41.786 --> 00:00:46.192
y is equal to the absolute
value of x minus negative three.

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And whatever you're
subtracting from this x,

00:00:48.608 --> 00:00:50.603
that is how much you are shifting it.

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So you're going to shift
it three to the left.

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And we're gonna do that right now

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and then we're gonna just gonna
confirm that it matches up.

00:00:57.997 --> 00:00:59.579
That it makes sense.

00:00:59.579 --> 00:01:01.897
So let's first graph that.

00:01:01.897 --> 00:01:03.163
Get my ruler tool here.

00:01:03.163 --> 00:01:04.864
So if we shift three to the left,

00:01:04.864 --> 00:01:07.959
it's gonna look something like...

00:01:07.959 --> 00:01:09.927
It's gonna look something like this.

00:01:09.927 --> 00:01:11.664
So on that...

00:01:11.664 --> 00:01:12.639
When whatever we have

00:01:12.639 --> 00:01:15.827
inside the absolute
value sign is positive,

00:01:15.827 --> 00:01:20.005
we're going to get
essentially, this slope of one.

00:01:20.005 --> 00:01:21.890
And whenever we have

00:01:21.890 --> 00:01:24.577
inside the absolute
value sign is negative,

00:01:24.577 --> 00:01:27.592
we're gonna have a slope of
essentially negative one.

00:01:27.592 --> 00:01:29.546
So it's going to look...

00:01:29.546 --> 00:01:32.216
It's going to look like that.

00:01:32.216 --> 00:01:36.023
And let's confirm whether
this actually makes sense.

00:01:36.023 --> 00:01:38.933
Below x equals negative three,

00:01:38.933 --> 00:01:41.558
for x values less than negative three,

00:01:41.558 --> 00:01:43.385
what we're gonna have here,

00:01:43.385 --> 00:01:46.477
is this inside of the absolute value sign,

00:01:46.477 --> 00:01:48.171
is going to be negative

00:01:48.171 --> 00:01:50.185
and so then we're gonna
take it's opposite value

00:01:50.185 --> 00:01:51.160
and so that makes sense.

00:01:51.160 --> 00:01:55.470
That's why you have this
downward line right over here.

00:01:55.470 --> 00:01:58.808
Now for x is greater than negative three,

00:01:58.808 --> 00:02:02.932
when you add three to it, you're
gonna get a positive value

00:02:02.932 --> 00:02:06.234
and so that's why you have
this upward sloping line

00:02:06.234 --> 00:02:07.388
right over here.

00:02:07.388 --> 00:02:09.436
And at x equals negative three,

00:02:09.436 --> 00:02:13.327
you have zero inside
the absolute value sign,

00:02:13.327 --> 00:02:15.375
just as if you didn't shift it,

00:02:15.375 --> 00:02:17.939
you would have had zero
inside the absolute value sign

00:02:17.939 --> 00:02:19.559
at x equals zero.

00:02:19.559 --> 00:02:22.121
So hopefully that makes a
little bit more intuitive sense

00:02:22.121 --> 00:02:24.795
of why if you replace x,

00:02:24.795 --> 00:02:27.867
if you replace x with x plus three,

00:02:27.867 --> 00:02:29.679
and this isn't just true of
absolute value functions,

00:02:29.679 --> 00:02:31.740
this is true of any function,

00:02:31.740 --> 00:02:34.138
if you replace x with x plus three,

00:02:34.138 --> 00:02:37.721
you are going to shift three to the left.

00:02:37.721 --> 00:02:39.789
All right, now let's keep building.

00:02:39.789 --> 00:02:41.981
Now let's see if we can graph

00:02:41.981 --> 00:02:46.148
y is equal to two times the
absolute value of x plus three.

00:02:47.377 --> 00:02:49.798
So what this is essentially going to do

00:02:49.798 --> 00:02:53.298
is multiple the slopes by a factor of two.

00:02:55.332 --> 00:02:59.836
It's going to stretch it
vertically by a factor of two.

00:02:59.836 --> 00:03:04.329
So for x values greater
than negative three,

00:03:04.329 --> 00:03:06.026
instead of having a slope of one,

00:03:06.026 --> 00:03:07.815
you're gonna have a slope of two.

00:03:07.815 --> 00:03:09.685
So let me get my ruler out again

00:03:09.685 --> 00:03:12.030
and see if I can draw that.

00:03:12.030 --> 00:03:14.113
So let me put that there.

00:03:15.590 --> 00:03:19.125
And then, so here instead
of a slope of one,

00:03:19.125 --> 00:03:22.758
I'm gonna have a slope of two.

00:03:22.758 --> 00:03:24.175
Let me draw that.

00:03:25.054 --> 00:03:27.382
It's gonna look like
that, right over there.

00:03:27.382 --> 00:03:31.023
And then instead of having
a slope of negative one

00:03:31.023 --> 00:03:34.182
for values less than x
equals negative three,

00:03:34.182 --> 00:03:37.432
I'm gonna have a slope of negative two.

00:03:38.459 --> 00:03:41.214
Let me draw that right over there.

00:03:41.214 --> 00:03:44.137
So that is the graph of y is equal to

00:03:44.137 --> 00:03:47.292
two times the absolute
value of x plus three.

00:03:47.292 --> 00:03:50.329
And now to get to the f
of x that we care about,

00:03:50.329 --> 00:03:52.810
we now need to add this two.

00:03:52.810 --> 00:03:56.839
So now I wanna graph, and
we're in the home stretch,

00:03:56.839 --> 00:03:59.921
I wanna graph, y is equal to two times

00:03:59.921 --> 00:04:04.153
the absolute value of
x plus three, plus two.

00:04:04.153 --> 00:04:06.080
Well whatever y value I was getting

00:04:06.080 --> 00:04:08.819
for this orange function,
I now wanna add two to it.

00:04:08.819 --> 00:04:12.104
So this is just gonna shift
it up vertically by two.

00:04:12.104 --> 00:04:13.437
So instead of...

00:04:14.662 --> 00:04:16.750
So this is gonna be shifted up by two.

00:04:16.750 --> 00:04:19.180
This is going to be shifted.

00:04:19.180 --> 00:04:23.042
Every point is going to shifted up by two

00:04:23.042 --> 00:04:26.241
or you can think about shifting
up the entire graph by two.

00:04:26.241 --> 00:04:27.601
Here, in the orange function,

00:04:27.601 --> 00:04:30.308
whatever y value I got
for the black function,

00:04:30.308 --> 00:04:32.591
I'm gonna have to get two more than that.

00:04:32.591 --> 00:04:34.924
And so it's going to look...

00:04:35.849 --> 00:04:38.623
It's going to look like this.

00:04:38.623 --> 00:04:41.845
So let me see, I'm shifting it up by two.

00:04:41.845 --> 00:04:45.078
So for x less than negative
three, it'll look like that.

00:04:45.078 --> 00:04:48.798
And for x greater than negative three,

00:04:48.798 --> 00:04:51.048
it is going to look like...

00:04:53.089 --> 00:04:55.589
It is going to look like that.

00:04:56.997 --> 00:04:58.212
And there you have it.

00:04:58.212 --> 00:05:02.358
This is the graph of y
or f of x is equal to

00:05:02.358 --> 00:05:05.098
two times the absolute value
of x plus three, plus two.

00:05:05.098 --> 00:05:06.669
And you could've done
it in different ways.

00:05:06.669 --> 00:05:09.442
You could have shifted up two first,

00:05:09.442 --> 00:05:14.274
then you could have
multiplied by a factor of two,

00:05:14.274 --> 00:05:17.282
and then you could have shifted, and then,

00:05:17.282 --> 00:05:18.929
so you could have moved up two first,

00:05:18.929 --> 00:05:20.797
then you coulda multiplied
by a factor of two,

00:05:20.797 --> 00:05:22.933
then you could've shifted left by three.

00:05:22.933 --> 00:05:24.570
Or could have multiplied
by a factor of two first,

00:05:24.570 --> 00:05:26.850
shifted up two and then shifted over.

00:05:26.850 --> 00:05:28.000
So there's multiple,

00:05:28.000 --> 00:05:29.770
there's three transformations
going up here.

00:05:29.770 --> 00:05:30.853
This is an...

00:05:31.812 --> 00:05:34.099
This is a, let me color them all.

00:05:34.099 --> 00:05:37.530
So this right over here tells me...

00:05:37.530 --> 00:05:41.026
This over here says hey, shift left.

00:05:41.026 --> 00:05:44.276
Shift left by three.

00:05:46.351 --> 00:05:49.684
This told us, stretch vertically by two.

00:05:51.884 --> 00:05:54.655
Or essentially multiply the slope by two.

00:05:54.655 --> 00:05:58.700
Stretch vert by two.

00:05:58.700 --> 00:06:00.440
And then that last piece,

00:06:00.440 --> 00:06:02.636
says shift up by two.

00:06:02.636 --> 00:06:05.303
Shift up by two,

00:06:06.360 --> 00:06:09.913
which gives us our
final result for f of x.


Scaling & reflecting absolute value functions: graph

WEBVTT
Kind: captions
Language: en

00:00:00.282 --> 00:00:02.090
- [Instructor] Function
G can be thought of

00:00:02.090 --> 00:00:05.027
as a stretched or compressed version

00:00:05.027 --> 00:00:08.144
of F of X is equal to
the absolute value of X.

00:00:08.144 --> 00:00:10.261
What is the equation for G of X?

00:00:10.261 --> 00:00:11.540
So you can see F of X is equal

00:00:11.540 --> 00:00:13.822
to the absolute value of X here in blue,

00:00:13.822 --> 00:00:15.290
and then G of X,

00:00:15.290 --> 00:00:17.102
not only does it look
stressed or compressed,

00:00:17.102 --> 00:00:20.613
but it also is flipped over the X axis.

00:00:20.613 --> 00:00:21.960
So like always, pause this video

00:00:21.960 --> 00:00:24.049
and see if you can up yourself

00:00:24.049 --> 00:00:26.772
with the equation for G of X.

00:00:26.772 --> 00:00:29.873
Alright, now let's work
through this together.

00:00:29.873 --> 00:00:31.753
So there's a couple of
ways we could do it.

00:00:31.753 --> 00:00:34.249
We could first try to flip F of X,

00:00:34.249 --> 00:00:35.677
and then try to stretch or compress it,

00:00:35.677 --> 00:00:37.419
or we could stretch or compress it first,

00:00:37.419 --> 00:00:39.258
and then try to flip it.

00:00:39.258 --> 00:00:41.285
Let's actually, let's flip it first,

00:00:41.285 --> 00:00:44.477
so let's say that we have a function

00:00:44.477 --> 00:00:45.310
that looks like this.

00:00:45.310 --> 00:00:47.607
It's just exactly what F of X is,

00:00:47.607 --> 00:00:50.568
but flipped over the X axis.

00:00:50.568 --> 00:00:53.023
So it's just flipped over the X axis,

00:00:53.023 --> 00:00:55.990
so all the values for any given X,

00:00:55.990 --> 00:00:57.708
whatever Y you used to get,

00:00:57.708 --> 00:01:00.104
you're not getting the negative of that.

00:01:00.104 --> 00:01:02.200
So this graph right over here,

00:01:02.200 --> 00:01:03.786
this would be the graph.

00:01:03.786 --> 00:01:06.758
I'll call this, Y is equal

00:01:06.758 --> 00:01:09.758
to the negative absolute value of X.

00:01:10.929 --> 00:01:12.592
Whatever the absolute value of X

00:01:12.592 --> 00:01:14.001
would have gotten you before,

00:01:14.001 --> 00:01:15.593
you're now going to get the negative

00:01:15.593 --> 00:01:17.288
of the opposite of it.

00:01:17.288 --> 00:01:20.214
So this is getting us
closer to our definition

00:01:20.214 --> 00:01:21.752
of G of X.

00:01:21.752 --> 00:01:25.152
The key here is how do
we appropriately stretch

00:01:25.152 --> 00:01:27.735
or squeeze this green function?

00:01:28.670 --> 00:01:30.609
So let's think about what's happening.

00:01:30.609 --> 00:01:31.917
On this green function,

00:01:31.917 --> 00:01:33.834
when X is equal to one,

00:01:34.881 --> 00:01:38.418
the function itself is
equal to negative one,

00:01:38.418 --> 00:01:40.827
but we want it, if we want
it to be the same as G,

00:01:40.827 --> 00:01:43.242
we want it to be equal to negative four.

00:01:43.242 --> 00:01:45.855
So it's actually four times the value.

00:01:45.855 --> 00:01:48.432
For a given X, at least for X equals one,

00:01:48.432 --> 00:01:51.163
G is giving me something
four times the value

00:01:51.163 --> 00:01:53.028
that my green function is giving.

00:01:53.028 --> 00:01:54.888
That's not just true for positive Xs.

00:01:54.888 --> 00:01:56.761
It's also true for negative Xs.

00:01:56.761 --> 00:01:57.775
You can see it right over here.

00:01:57.775 --> 00:02:00.830
When X is equal to negative one,

00:02:00.830 --> 00:02:04.429
my green function gives me negative one,

00:02:04.429 --> 00:02:07.273
but G gives me negative four.

00:02:07.273 --> 00:02:09.158
So it's giving me four times the value.

00:02:09.158 --> 00:02:10.628
It's giving me four
times the negative value,

00:02:10.628 --> 00:02:12.904
so it's going even more negative,

00:02:12.904 --> 00:02:15.220
so what you can see, to
go from the green to G,

00:02:15.220 --> 00:02:20.148
you have to multiply this
thing right over here by four.

00:02:20.148 --> 00:02:22.732
So that is what essentially
stretches it down,

00:02:22.732 --> 00:02:25.216
stretches it down in
the vertical direction.

00:02:25.216 --> 00:02:29.337
So we could say that G of X is equal to,

00:02:29.337 --> 00:02:31.381
it's not negative absolute value of X,

00:02:31.381 --> 00:02:35.048
negative four times the
absolute value of X.

00:02:36.089 --> 00:02:37.430
And you could have done it the other way.

00:02:37.430 --> 00:02:39.681
You could have said,
"Hey, let's first stretch

00:02:39.681 --> 00:02:41.023
"or compress F."

00:02:41.023 --> 00:02:43.223
And say, alright, before
we even flip it over,

00:02:43.223 --> 00:02:44.656
if we were to unflip G,

00:02:44.656 --> 00:02:46.434
it would look like this.

00:02:46.434 --> 00:02:48.220
If we were to unflip G,

00:02:48.220 --> 00:02:50.220
it would look like this.

00:02:53.428 --> 00:02:55.095
If were to unflip G,

00:02:56.298 --> 00:02:58.131
so this thing right over here,

00:02:58.131 --> 00:03:01.434
this thing looks like four times F of X.

00:03:01.434 --> 00:03:03.267
We could write this as

00:03:05.293 --> 00:03:07.960
Y is equal to four times F of X,

00:03:09.388 --> 00:03:12.841
or you could say Y is equal to four times

00:03:12.841 --> 00:03:14.729
the absolute value of X,

00:03:14.729 --> 00:03:16.656
and then we have a negative sign.

00:03:16.656 --> 00:03:18.645
Whatever positive value
you were getting before,

00:03:18.645 --> 00:03:20.659
you now get the opposite value,

00:03:20.659 --> 00:03:22.698
and that would flip it
over and get you to G,

00:03:22.698 --> 00:03:25.865
which is exactly what we already got.


Flipping and compressing a graph

WEBVTT
Kind: captions
Language: en

00:00:00.906 --> 00:00:02.675
- [Instructor] The graph of y is equal to

00:00:02.675 --> 00:00:07.124
the absolute value of x is
reflected across the x-axis

00:00:07.124 --> 00:00:11.291
and then compressed
vertically by a factor of 8/3.

00:00:12.281 --> 00:00:15.521
What is the equation of the new graph?

00:00:15.521 --> 00:00:18.769
All right, so let's think
about this step-by-step.

00:00:18.769 --> 00:00:20.835
If I start, and I'm just going to draw

00:00:20.835 --> 00:00:24.109
some quick hand-drawn sketches here.

00:00:24.109 --> 00:00:25.866
So that's my x-axis.

00:00:25.866 --> 00:00:27.366
That is my y-axis.

00:00:28.233 --> 00:00:31.606
If we're talking about the graph of

00:00:31.606 --> 00:00:34.077
y is equal to the absolute value of x,

00:00:34.077 --> 00:00:36.415
that looks like this.

00:00:36.415 --> 00:00:39.371
In the first quadrant,
it looks like y equals x,

00:00:39.371 --> 00:00:41.393
and in the second quadrant,

00:00:41.393 --> 00:00:44.668
it looks like y equals negative x,

00:00:44.668 --> 00:00:46.557
'cause the absolute value
of a negative number

00:00:46.557 --> 00:00:48.196
is its opposite.

00:00:48.196 --> 00:00:49.449
So let me make it look a little bit more

00:00:49.449 --> 00:00:51.163
symmetric than that.

00:00:51.163 --> 00:00:53.289
So it looks something,

00:00:53.289 --> 00:00:55.468
something like that.

00:00:55.468 --> 00:00:57.909
That's what the graph of y is equal to the

00:00:57.909 --> 00:01:00.709
absolute value of x looks like.

00:01:00.709 --> 00:01:03.326
Now, they're asking us
to do what amounts to

00:01:03.326 --> 00:01:05.852
two different transformations.

00:01:05.852 --> 00:01:09.269
The first one is they want to reflect it.

00:01:10.476 --> 00:01:13.863
They want to reflect across the x-axis,

00:01:13.863 --> 00:01:17.853
so they want us to flip it
across the x-axis like this.

00:01:17.853 --> 00:01:21.173
So instead, it looks like this.

00:01:21.173 --> 00:01:25.340
So that graph, that equation
that describes this graph,

00:01:26.350 --> 00:01:28.955
well, this is going to be the opposite.

00:01:28.955 --> 00:01:30.934
Whatever y you were getting
on this orange graph,

00:01:30.934 --> 00:01:32.882
you're going to get the negative of that.

00:01:32.882 --> 00:01:35.141
You're going to get y
is equal to the negative

00:01:35.141 --> 00:01:37.555
of the absolute value of x.

00:01:37.555 --> 00:01:40.756
If this doesn't make intuitive
sense to you, try it out.

00:01:40.756 --> 00:01:44.923
In the orange graph, when x
is equal to, let's say, two,

00:01:46.647 --> 00:01:49.206
well the absolute value of two is two.

00:01:49.206 --> 00:01:52.343
The absolute value of negative two is two.

00:01:52.343 --> 00:01:54.974
Now, we want to take the absolute value

00:01:54.974 --> 00:01:56.533
but then take the negative of it.

00:01:56.533 --> 00:01:59.501
This thing stays
non-positive the entire time,

00:01:59.501 --> 00:02:01.593
so the absolute value of two is two,

00:02:01.593 --> 00:02:03.566
but we want the opposite of that.

00:02:03.566 --> 00:02:04.914
We want negative two.

00:02:04.914 --> 00:02:07.111
Absolute value of negative two is two,

00:02:07.111 --> 00:02:10.085
but we want the opposite of that.

00:02:10.085 --> 00:02:13.326
They didn't ask us to just
reflect across the x-axis.

00:02:13.326 --> 00:02:16.647
They then want us to compress vertically

00:02:16.647 --> 00:02:18.993
by a factor of 8/3.

00:02:18.993 --> 00:02:20.706
So let's think about this a little bit.

00:02:20.706 --> 00:02:24.465
Compress vertically by a factor of 8/3.

00:02:24.465 --> 00:02:27.835
If they said stretch
vertically by a factor of 8/3,

00:02:27.835 --> 00:02:30.752
then I would just multiply this by 8/3.

00:02:30.752 --> 00:02:32.118
Sometimes it's helpful to think of this

00:02:32.118 --> 00:02:33.613
in terms of a mixed number.

00:02:33.613 --> 00:02:37.941
8/3 is the same thing as 2 2/3.

00:02:37.941 --> 00:02:42.108
If you were to stretch by 2
2/3, you would get taller.

00:02:45.002 --> 00:02:47.327
It would get something like this.

00:02:47.327 --> 00:02:49.181
If you are compressing,

00:02:49.181 --> 00:02:51.672
then it's going to look
something like this.

00:02:51.672 --> 00:02:53.140
If you're compressing vertically,

00:02:53.140 --> 00:02:56.805
you can think about it as
being stretched horizontally.

00:02:56.805 --> 00:02:59.671
If you're compressing by a factor,

00:02:59.671 --> 00:03:02.492
you should multiply by the
reciprocal of that factor.

00:03:02.492 --> 00:03:04.962
Think about it, if you
were compressing by three,

00:03:04.962 --> 00:03:07.994
you would multiply by one-third.

00:03:07.994 --> 00:03:11.433
If you were compressing vertically by 8/3,

00:03:11.433 --> 00:03:14.352
that means whatever y
you would have gotten,

00:03:14.352 --> 00:03:17.442
you multiply that times
a reciprocal of 8/3.

00:03:17.442 --> 00:03:18.848
Another way you could think about it,

00:03:18.848 --> 00:03:21.623
you could divide that by 8/3.

00:03:21.623 --> 00:03:25.373
If we want to get this
thing right over here,

00:03:26.812 --> 00:03:30.060
this is going to be y is equal to

00:03:30.060 --> 00:03:32.873
the negative absolute value of x.

00:03:32.873 --> 00:03:35.033
Since we're compressing by 8/3,

00:03:35.033 --> 00:03:37.584
we would divide that by 8/3.

00:03:37.584 --> 00:03:39.206
Another way to think about it,

00:03:39.206 --> 00:03:40.591
dividing by 8/34 is the same thing

00:03:40.591 --> 00:03:42.413
as multiplying by the reciprocal.

00:03:42.413 --> 00:03:47.340
So negative 3/8 times
the absolute value of x.

00:03:47.340 --> 00:03:51.621
When we look at our
choices, we see that it is

00:03:51.621 --> 00:03:54.182
that choice right over there.

00:03:54.182 --> 00:03:55.508
I really want to stress this point,

00:03:55.508 --> 00:03:58.034
because I think it can get
a little bit confusing.

00:03:58.034 --> 00:04:01.261
The reason why I multiplied
that by the reciprocal,

00:04:01.261 --> 00:04:04.060
they were saying compressing vertically.

00:04:04.060 --> 00:04:05.343
If we said stretching vertically,

00:04:05.343 --> 00:04:07.222
we would just multiply by 8/3,

00:04:07.222 --> 00:04:08.433
but since we're compressing,

00:04:08.433 --> 00:04:12.600
we would divide by 8/3 or
multiply by its reciprocal.


Shifting absolute value graphs

WEBVTT
Kind: captions
Language: en

00:00:00.003 --> 00:00:01.043
- [Instructor] This right over here

00:00:01.043 --> 00:00:04.607
is the graph of y is equal
to absolute value of x

00:00:04.607 --> 00:00:06.037
which you might be familiar with.

00:00:06.037 --> 00:00:08.213
If you take x is equal to negative two,

00:00:08.213 --> 00:00:11.431
the absolute value of
that is going to be two.

00:00:11.431 --> 00:00:13.473
Negative one, absolute value is one.

00:00:13.473 --> 00:00:15.308
Zero, absolute value is zero.

00:00:15.308 --> 00:00:17.735
One, absolute value is one.

00:00:17.735 --> 00:00:19.514
So on and so forth.

00:00:19.514 --> 00:00:20.786
What I wanna do in this video

00:00:20.786 --> 00:00:23.992
is think about how the
equation will change

00:00:23.992 --> 00:00:26.327
if we were to shift this graph.

00:00:26.327 --> 00:00:28.457
So in particular, we're
gonna first think about

00:00:28.457 --> 00:00:32.624
what would be the equation
of this graph if we shift,

00:00:33.633 --> 00:00:36.133
if we shift three to the right

00:00:38.132 --> 00:00:41.034
and then think about how that will change

00:00:41.034 --> 00:00:43.038
if not only do we shift three to the right

00:00:43.038 --> 00:00:45.205
but we also shift four up,

00:00:46.869 --> 00:00:48.036
shift four up,

00:00:49.855 --> 00:00:53.617
and so once again pause this
video like we always say

00:00:53.617 --> 00:00:55.785
and figure out what would the equation be

00:00:55.785 --> 00:00:59.674
if you shift three to
the right and four up?

00:00:59.674 --> 00:01:01.347
Alright, now let's do this together.

00:01:01.347 --> 00:01:03.512
So let's just first
shift three to the right

00:01:03.512 --> 00:01:06.339
and think about how that
might change the equation.

00:01:06.339 --> 00:01:10.955
So let's just visualize what
we're even talking about.

00:01:10.955 --> 00:01:14.255
So if we're gonna shift
three to the right,

00:01:14.255 --> 00:01:16.290
it would look like,

00:01:16.290 --> 00:01:18.781
it would look like this.

00:01:18.781 --> 00:01:22.948
So that's what we first wanna
figure out the equation for

00:01:24.396 --> 00:01:27.636
and so how would we think about it?

00:01:27.636 --> 00:01:29.812
Well, one way to think about it is,

00:01:29.812 --> 00:01:31.783
well, something interesting is happening

00:01:31.783 --> 00:01:34.275
right over here at x equals three.

00:01:34.275 --> 00:01:36.317
Before, that interesting
thing was happening

00:01:36.317 --> 00:01:37.819
at x equals zero.

00:01:37.819 --> 00:01:40.168
Now, it's happening at x equals three.

00:01:40.168 --> 00:01:41.928
And the interesting
thing that happens here

00:01:41.928 --> 00:01:45.279
is that you switch signs
inside the absolute value.

00:01:45.279 --> 00:01:47.818
Instead of taking an
absolute value of a negative,

00:01:47.818 --> 00:01:49.454
you're now taking the absolute value

00:01:49.454 --> 00:01:52.191
as you cross this point of a positive

00:01:52.191 --> 00:01:55.507
and that's why we see a
switch in direction here

00:01:55.507 --> 00:01:56.507
of this line

00:01:57.591 --> 00:02:00.576
and so you see the same thing
happening right over here.

00:02:00.576 --> 00:02:03.326
So at this point right over here,

00:02:04.412 --> 00:02:07.105
we know that our function,

00:02:07.105 --> 00:02:11.560
we know that our equation
needs to evaluate out to zero

00:02:11.560 --> 00:02:15.227
and this is where it's
going to switch signs

00:02:16.330 --> 00:02:19.235
and so we say, okay, well, this
looks like an absolute value

00:02:19.235 --> 00:02:20.750
so it's going to have the form,

00:02:20.750 --> 00:02:25.330
y is equal to the absolute
value of something

00:02:25.330 --> 00:02:28.109
and so you say, okay, if x is three,

00:02:28.109 --> 00:02:31.265
how do I make that equal to zero?

00:02:31.265 --> 00:02:34.571
Well, I can subtract three from it.

00:02:34.571 --> 00:02:38.738
If I say y is equal to the
absolute value of x minus three,

00:02:39.761 --> 00:02:40.623
well, let's try it out.

00:02:40.623 --> 00:02:42.426
Let's see if it makes sense.

00:02:42.426 --> 00:02:44.597
So when x is equal to three,

00:02:44.597 --> 00:02:46.316
three minus three is zero,

00:02:46.316 --> 00:02:47.666
absolute value of that is zero.

00:02:47.666 --> 00:02:48.853
That works out.

00:02:48.853 --> 00:02:50.445
When x is equal to four,

00:02:50.445 --> 00:02:54.612
four minus three is one,
absolute value of one is indeed,

00:02:55.794 --> 00:02:56.961
is indeed one.

00:02:58.000 --> 00:02:59.962
And if x is equal to two,

00:02:59.962 --> 00:03:02.215
well, two minus three is negative one

00:03:02.215 --> 00:03:04.791
but the absolute value of that is one.

00:03:04.791 --> 00:03:06.586
So once again, I'm showing you this

00:03:06.586 --> 00:03:08.011
by really trying out numbers,

00:03:08.011 --> 00:03:09.793
trying to give you a
little bit of an intuition

00:03:09.793 --> 00:03:12.422
because that wasn't obvious to
me when I first learned this

00:03:12.422 --> 00:03:14.914
that if I'm shifting to the right

00:03:14.914 --> 00:03:17.528
which it looks like I'm
increasing an x value

00:03:17.528 --> 00:03:20.516
but what I would really do is replace my x

00:03:20.516 --> 00:03:24.520
with an x minus the amount
that I'm shifting to the right

00:03:24.520 --> 00:03:26.072
but I encourage you to try numbers

00:03:26.072 --> 00:03:28.086
and think about what's happening here.

00:03:28.086 --> 00:03:30.669
At this vertex right over here,

00:03:32.447 --> 00:03:36.737
whatever was in the absolute
value sign was equaling zero.

00:03:36.737 --> 00:03:38.660
It's when whatever was in
the absolute value sign

00:03:38.660 --> 00:03:41.851
is switching from negative
signs to positive signs.

00:03:41.851 --> 00:03:45.165
So once again, if you
shift three to the right,

00:03:45.165 --> 00:03:47.899
that has to happen at x equals three.

00:03:47.899 --> 00:03:51.332
So whatever is inside
the absolute value sign

00:03:51.332 --> 00:03:55.263
has to be equal to zero at
x equals three and this,

00:03:55.263 --> 00:03:57.050
pause this video and
really think about this

00:03:57.050 --> 00:03:58.569
if it isn't making sense

00:03:58.569 --> 00:04:01.345
and even as you get more
and more familiar with this,

00:04:01.345 --> 00:04:02.897
I encourage you to try out the numbers.

00:04:02.897 --> 00:04:05.427
That will give you more,
instead of just memorizing,

00:04:05.427 --> 00:04:07.103
hey, if I shift to the right,

00:04:07.103 --> 00:04:10.335
I replace x with x minus
the amount that I shift.

00:04:10.335 --> 00:04:12.340
Always try out the numbers
and try to get an intuition

00:04:12.340 --> 00:04:15.078
for that why that works.

00:04:15.078 --> 00:04:18.588
But now, let's start from
here and shift four up

00:04:18.588 --> 00:04:22.708
and shifting four up is in
some ways a lot more intuitive.

00:04:22.708 --> 00:04:24.208
So let me do that.

00:04:25.322 --> 00:04:27.072
Let me shift four up.

00:04:29.485 --> 00:04:31.485
Alright, so let me move.

00:04:33.286 --> 00:04:36.703
I'm gonna go up one, two, three and four.

00:04:39.501 --> 00:04:41.180
I think I got that right.

00:04:41.180 --> 00:04:44.340
So now I've shifted four up.

00:04:44.340 --> 00:04:46.975
And just as a reminder
of what we've even done,

00:04:46.975 --> 00:04:50.642
the first part we shifted
three to the right

00:04:51.508 --> 00:04:54.175
and now we are shifting four up.

00:04:55.553 --> 00:04:58.553
So now, now we are shifting four up.

00:05:01.313 --> 00:05:03.813
So before, y equaled zero here

00:05:05.068 --> 00:05:07.886
but now, y needs to be equal to four.

00:05:07.886 --> 00:05:09.552
So whatever this was evaluating,

00:05:09.552 --> 00:05:11.886
do we now have to add four to it?

00:05:11.886 --> 00:05:14.015
So when we just shifted
three to the right,

00:05:14.015 --> 00:05:17.123
our equation was y is
equal to the absolute value

00:05:17.123 --> 00:05:19.085
of x minus three

00:05:19.085 --> 00:05:20.847
but now whatever we were getting before,

00:05:20.847 --> 00:05:22.279
we now have to add four to it.

00:05:22.279 --> 00:05:25.386
We're going up in the vertical direction.

00:05:25.386 --> 00:05:27.719
So we just have to add four.

00:05:28.862 --> 00:05:30.659
Now, this makes a little bit more sense.

00:05:30.659 --> 00:05:33.720
If you're shifting in
the vertical direction,

00:05:33.720 --> 00:05:35.598
if you shift up in the vertical direction,

00:05:35.598 --> 00:05:38.828
well, you just add a constant
by the amount you're shifting.

00:05:38.828 --> 00:05:40.908
If you shift down in
the vertical direction,

00:05:40.908 --> 00:05:41.894
well, you would subtract.

00:05:41.894 --> 00:05:43.451
If we said shift down four,

00:05:43.451 --> 00:05:46.027
you would subtract four right over here.

00:05:46.027 --> 00:05:48.444
The less intuitive thing
is what we did with x

00:05:48.444 --> 00:05:50.205
'cause when you shift to the right,

00:05:50.205 --> 00:05:52.090
you actually replace your x

00:05:52.090 --> 00:05:55.174
with x minus the amount that you shifted

00:05:55.174 --> 00:05:57.197
but once again, try out numbers

00:05:57.197 --> 00:06:00.631
until it really makes some
intuitive sense for you

00:06:00.631 --> 00:06:02.636
but this is what we would finally get.

00:06:02.636 --> 00:06:05.583
The equation of this thing right over here

00:06:05.583 --> 00:06:07.997
is y is equal to the absolute value of x

00:06:07.997 --> 00:06:09.830
minus three plus four.


Analyzing structure with linear inequalities: fruits

WEBVTT
Kind: captions
Language: en

00:00:00.321 --> 00:00:03.295
- [Instructor] Shantanu bought
more apples than bananas

00:00:03.295 --> 00:00:06.458
and he bought more
bananas than cantaloupes.

00:00:06.458 --> 00:00:09.603
Let A represent the number
of apples Shantanu bought,

00:00:09.603 --> 00:00:11.635
let B represent the number of bananas,

00:00:11.635 --> 00:00:14.353
and let C represent the
number of cantaloupes.

00:00:14.353 --> 00:00:18.774
Let's compare the
expressions B plus C and A.

00:00:18.774 --> 00:00:20.791
Which statement is correct?

00:00:20.791 --> 00:00:23.078
So is B plus C greater than A?

00:00:23.078 --> 00:00:24.331
Is it less than A?

00:00:24.331 --> 00:00:26.225
Or are these two quantities equal?

00:00:26.225 --> 00:00:28.612
Or is there not enough
information to tell?

00:00:28.612 --> 00:00:30.755
So like always, pause this video

00:00:30.755 --> 00:00:32.403
and see if you can work
through it on your own

00:00:32.403 --> 00:00:35.537
and now I will work through it with you.

00:00:35.537 --> 00:00:37.416
All right, so let's just
write down the information

00:00:37.416 --> 00:00:38.467
that they gave us.

00:00:38.467 --> 00:00:40.470
They say let A represent the number,

00:00:40.470 --> 00:00:41.791
well that's more straightforward,

00:00:41.791 --> 00:00:44.229
A for apple, B for
banana, C for cantaloupe.

00:00:44.229 --> 00:00:47.494
Here we have more apples than bananas,

00:00:47.494 --> 00:00:49.411
so A is greater than B,

00:00:51.164 --> 00:00:54.349
and then they also tell
us he bought more bananas

00:00:54.349 --> 00:00:57.766
than cantaloupes, so B is greater than C,

00:00:59.269 --> 00:01:01.671
or we could rewrite that as

00:01:01.671 --> 00:01:04.754
A is greater than B is greater than C

00:01:06.717 --> 00:01:10.884
or we could write that as C
is less than B is less than A.

00:01:12.399 --> 00:01:15.991
This is essentially the
information that they give us.

00:01:15.991 --> 00:01:19.510
So let's see, which of
these is going to be true?

00:01:19.510 --> 00:01:21.510
B plus C greater than A,

00:01:23.471 --> 00:01:25.866
B plus C greater than A,

00:01:25.866 --> 00:01:27.616
B plus C less than A.

00:01:28.917 --> 00:01:33.358
So one thing that we can
try is let's try to plug in

00:01:33.358 --> 00:01:35.519
some values, some numbers

00:01:35.519 --> 00:01:37.173
to see if we can get combinations

00:01:37.173 --> 00:01:39.261
that are consistently
in one of these buckets

00:01:39.261 --> 00:01:41.114
or if they fall into
multiple of these choices,

00:01:41.114 --> 00:01:43.596
then we say hey there's not
enough information to tell.

00:01:43.596 --> 00:01:46.126
In general, this is a good
strategy for things like this

00:01:46.126 --> 00:01:49.009
where we're dealing with
very abstract quantities.

00:01:49.009 --> 00:01:51.795
So let's make a little table here.

00:01:51.795 --> 00:01:52.628
So A, B, C

00:01:55.038 --> 00:01:58.817
and then I can also figure
out what B plus C is.

00:01:58.817 --> 00:02:02.005
So this is going to be A, B, C,

00:02:02.005 --> 00:02:05.739
and this is B plus C and
we can compare that to A.

00:02:05.739 --> 00:02:07.506
So let's see a situation where

00:02:07.506 --> 00:02:11.589
let's see if we can make
B plus C greater than A.

00:02:13.274 --> 00:02:15.756
So they both have to be less than A.

00:02:15.756 --> 00:02:17.923
So let's see, if C is five

00:02:23.650 --> 00:02:24.650
and B is six

00:02:26.473 --> 00:02:28.390
and let's make A seven.

00:02:30.724 --> 00:02:34.956
So in this situation, B plus
C is going to be equal to 11.

00:02:34.956 --> 00:02:37.024
So we're able to find a situation

00:02:37.024 --> 00:02:39.934
where if B plus C are close enough to A,

00:02:39.934 --> 00:02:41.922
that B plus C is going
to be greater than A.

00:02:41.922 --> 00:02:44.650
So we're able to find this scenario.

00:02:44.650 --> 00:02:46.196
Let's see if we can figure out a scenario

00:02:46.196 --> 00:02:48.696
where B plus C is less than A.

00:02:50.114 --> 00:02:50.947
Well if,

00:02:52.041 --> 00:02:56.050
well we could do the
same B plus C, six, five.

00:02:56.050 --> 00:02:58.379
We could make A bigger than six plus five.

00:02:58.379 --> 00:02:59.796
We can make A 12.

00:03:00.698 --> 00:03:02.181
And now this is a situation,

00:03:02.181 --> 00:03:06.433
so this first situation we have
B plus C is greater than A.

00:03:06.433 --> 00:03:08.023
The second situation right over here,

00:03:08.023 --> 00:03:10.690
you have B plus C is less than A

00:03:11.738 --> 00:03:14.666
and so depending on what
your A, B, and C's are

00:03:14.666 --> 00:03:16.200
that meet these constraints and notice,

00:03:16.200 --> 00:03:18.388
both of these situations
I meet all the constraints

00:03:18.388 --> 00:03:20.917
where A is greater than
B is greater than C,

00:03:20.917 --> 00:03:22.365
but it could be either one of these.

00:03:22.365 --> 00:03:23.877
So that immediately tells us

00:03:23.877 --> 00:03:27.627
that there is not enough
information to tell.

00:03:29.320 --> 00:03:31.757
Now, one thing that we,

00:03:31.757 --> 00:03:34.341
yeah there's just not
enough information to tell.

00:03:34.341 --> 00:03:36.217
I can even come up with a scenario

00:03:36.217 --> 00:03:37.998
where B plus C is equal to A.

00:03:37.998 --> 00:03:40.165
If it's six, five, and 11.

00:03:41.494 --> 00:03:43.827
Then B plus C is equal to A.

00:03:44.968 --> 00:03:46.669
So based on the information they gave us,

00:03:46.669 --> 00:03:48.831
any of these are actually possible.

00:03:48.831 --> 00:03:51.917
So there's not enough information to tell.


Analyzing structure with linear inequalities: balls

WEBVTT
Kind: captions
Language: en

00:00:00.324 --> 00:00:03.670
- [Instructor] A bag has more
green balls than blue balls,

00:00:03.670 --> 00:00:06.175
and there is at least one blue ball.

00:00:06.175 --> 00:00:08.721
Let b represent the number of blue balls

00:00:08.721 --> 00:00:12.109
and let g represent the
number of green balls.

00:00:12.109 --> 00:00:16.234
Let's compare the
expressions 2b and b plus g.

00:00:16.234 --> 00:00:18.079
Which statement is correct?

00:00:18.079 --> 00:00:21.554
So they make a bunch of comparisons
between 2b and b plus g.

00:00:21.554 --> 00:00:23.269
Is it greater than, less than, equal,

00:00:23.269 --> 00:00:25.783
or is there just not
enough information to tell?

00:00:25.783 --> 00:00:27.578
And so like always, pause this video

00:00:27.578 --> 00:00:29.581
and see if you can
figure it out on your own

00:00:29.581 --> 00:00:31.551
before we work through it together.

00:00:31.551 --> 00:00:32.713
Alright, let's work through it together,

00:00:32.713 --> 00:00:35.454
and let's see what information
they are giving us.

00:00:35.454 --> 00:00:37.865
So this first sentence they say,

00:00:37.865 --> 00:00:41.610
a bag has more green
balls than blue balls.

00:00:41.610 --> 00:00:44.896
So we can translate that in math language

00:00:44.896 --> 00:00:46.695
as the number of green balls

00:00:46.695 --> 00:00:50.748
is greater than the number of blue balls.

00:00:50.748 --> 00:00:54.425
It also says, there is
at least one blue ball.

00:00:54.425 --> 00:00:56.012
So just translating that mathematically,

00:00:56.012 --> 00:01:00.486
we know that b is going to be
greater than or equal to one.

00:01:00.486 --> 00:01:02.941
There is at least one, but maybe more.

00:01:02.941 --> 00:01:05.503
So b is greater than or equal to one.

00:01:05.503 --> 00:01:09.028
So now let's see if we can
somehow manipulate these

00:01:09.028 --> 00:01:11.650
so we can get to some reasonable sense

00:01:11.650 --> 00:01:14.233
of how b plus g compares to 2b.

00:01:15.984 --> 00:01:17.783
Well, there's a bunch of different ways

00:01:17.783 --> 00:01:18.763
that you could approach it.

00:01:18.763 --> 00:01:20.050
And you might find a way to approach it

00:01:20.050 --> 00:01:22.828
that is different than my way.

00:01:22.828 --> 00:01:24.634
But the one that jumps out at me is,

00:01:24.634 --> 00:01:27.273
well, I know that g is greater than b,

00:01:27.273 --> 00:01:29.456
and if I add a b to both sides,

00:01:29.456 --> 00:01:32.605
on the left-hand side I'm
gonna have a b plus g,

00:01:32.605 --> 00:01:35.305
and on the right-hand side
I'm gonna have a b plus b.

00:01:35.305 --> 00:01:38.134
And this right over here is 2b.

00:01:38.134 --> 00:01:41.272
So just like that, I know
I can keep the inequality

00:01:41.272 --> 00:01:44.199
if I'm doing the same thing to both sides,

00:01:44.199 --> 00:01:47.653
if I'm adding or subtracting
the same thing to both sides.

00:01:47.653 --> 00:01:50.559
And so just from g is greater than b,

00:01:50.559 --> 00:01:52.698
if you add b to both sides of that,

00:01:52.698 --> 00:01:54.948
we can deduce that b plus g

00:01:56.288 --> 00:01:59.124
is going to be greater than b plus b,

00:01:59.124 --> 00:02:02.080
or it's just going to be greater than 2b.

00:02:02.080 --> 00:02:03.339
And that's all we need to do.

00:02:03.339 --> 00:02:05.810
And let's see which of these
choices match up to that.

00:02:05.810 --> 00:02:08.227
So let's see, all of these
have 2b on the left-hand side.

00:02:08.227 --> 00:02:13.149
So we could rewrite this as
2b is less than b plus g,

00:02:13.149 --> 00:02:16.566
and that is that choice right over there.


Graphing two variable inequality

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.096
- [Instructor] So what I
would like to do in this video

00:00:02.096 --> 00:00:05.179
is graph the inequality negative 14 X

00:00:06.527 --> 00:00:09.194
minus seven Y is less than four.

00:00:10.166 --> 00:00:12.205
Like always, I encourage
you to pause this video

00:00:12.205 --> 00:00:13.457
and see if you can graph it on your own

00:00:13.457 --> 00:00:16.298
before we work through it together.

00:00:16.298 --> 00:00:18.578
So the way that I like to do this

00:00:18.578 --> 00:00:23.194
is I like to isolate a Y on
one side of this inequality,

00:00:23.194 --> 00:00:25.864
and I'll do it on the left-hand side.

00:00:25.864 --> 00:00:27.426
So the first thing I want
to do is let's get rid

00:00:27.426 --> 00:00:30.021
of this negative 14 X here on the left.

00:00:30.021 --> 00:00:32.784
So the best way to do that is to add 14 X,

00:00:32.784 --> 00:00:34.705
but if I do that to the left-hand side

00:00:34.705 --> 00:00:38.246
I also want to do that on
the right-hand side as well.

00:00:38.246 --> 00:00:41.613
So I'm adding 14 X to both sides,

00:00:41.613 --> 00:00:44.493
and that leaves me with
these two cancel out.

00:00:44.493 --> 00:00:47.645
On the left-hand side I
have negative seven Y,

00:00:47.645 --> 00:00:52.043
and that's going to be less
than, I have 14 X plus four,

00:00:52.043 --> 00:00:54.035
or four plus 14 X.

00:00:54.035 --> 00:00:55.285
14 X plus four.

00:00:57.401 --> 00:01:01.779
And now I want to divide
both sides by negative seven

00:01:01.779 --> 00:01:03.908
in order to isolate this Y.

00:01:03.908 --> 00:01:06.377
But we have to be very,
very, very careful.

00:01:06.377 --> 00:01:09.986
When you multiply or divide
both sides of an inequality

00:01:09.986 --> 00:01:13.576
by a negative number,
you need to swap this,

00:01:13.576 --> 00:01:15.680
you need to swap the inequality sign.

00:01:15.680 --> 00:01:16.853
Instead of it being less than,

00:01:16.853 --> 00:01:18.562
it would have to become greater than.

00:01:18.562 --> 00:01:22.121
So if you divide both
sides by negative seven,

00:01:22.121 --> 00:01:26.121
you are going to be left
with Y is greater than,

00:01:27.020 --> 00:01:28.052
let me do that in a different color

00:01:28.052 --> 00:01:29.998
just to highlight that
I'm swapping this around.

00:01:29.998 --> 00:01:34.165
Y is gonna be greater than
14 divided by negative seven

00:01:35.951 --> 00:01:37.451
is negative two X.

00:01:38.797 --> 00:01:42.234
And then four divided by negative seven,

00:01:42.234 --> 00:01:45.234
well that's negative 4/7.

00:01:47.018 --> 00:01:49.439
And now I'm ready to graph this.

00:01:49.439 --> 00:01:50.843
And the way that I would graph this,

00:01:50.843 --> 00:01:55.010
I would graph Y is equal to
negative two X minus 4/7.

00:01:58.181 --> 00:02:00.750
But since we're not greater
than or equal to it,

00:02:00.750 --> 00:02:02.517
I would put a dotted line there

00:02:02.517 --> 00:02:04.595
and just shade everything above that.

00:02:04.595 --> 00:02:07.289
But let me just do what I'm talking about.

00:02:07.289 --> 00:02:09.789
So let me draw some axes here.

00:02:12.061 --> 00:02:14.005
So this is gonna be an
approximation obviously.

00:02:14.005 --> 00:02:17.638
This is a hand-drawn graph
that we're dealing with.

00:02:17.638 --> 00:02:20.638
So it's like that, that's my x-axis.

00:02:21.619 --> 00:02:24.667
And let's see, our
y-intercept is negative 4/7,

00:02:24.667 --> 00:02:26.591
which is a little bit more than 1/2.

00:02:26.591 --> 00:02:29.892
So let's say that this right over here is,

00:02:29.892 --> 00:02:32.631
let's say this right over here is one,

00:02:32.631 --> 00:02:34.172
or negative one, I should say.

00:02:34.172 --> 00:02:36.151
Actually, I think I'm
gonna have to put my axis

00:02:36.151 --> 00:02:38.036
a little bit higher.

00:02:38.036 --> 00:02:40.064
Let's do it like this.

00:02:40.064 --> 00:02:41.509
So that is X.

00:02:41.509 --> 00:02:43.818
Let's make this negative one.

00:02:43.818 --> 00:02:46.960
And then that's about negative two.

00:02:46.960 --> 00:02:49.675
And so this would be one,

00:02:49.675 --> 00:02:53.092
and then this is going to be right about,

00:02:54.093 --> 00:02:58.260
this is about one, and then
we have two right over there.

00:02:59.551 --> 00:03:03.086
So we look at our y-intercept, when X is,

00:03:03.086 --> 00:03:05.217
so let's just think
about what would we have

00:03:05.217 --> 00:03:06.755
if we're thinking about Y is equal

00:03:06.755 --> 00:03:10.639
to negative two X minus 4/7.

00:03:10.639 --> 00:03:15.143
Well our y-intercept right over
here would be negative 4/7,

00:03:15.143 --> 00:03:17.981
which is a little bit more than 1/2,

00:03:17.981 --> 00:03:21.028
so it'll be roughly right around there.

00:03:21.028 --> 00:03:24.513
And then we have a slope of negative two.

00:03:24.513 --> 00:03:27.187
We have a slope of negative two.

00:03:27.187 --> 00:03:29.354
So if you increase by one,

00:03:30.738 --> 00:03:34.235
you are going to decrease by two.

00:03:34.235 --> 00:03:36.932
So you're gonna decrease,
you're not gonna just go there,

00:03:36.932 --> 00:03:39.979
you're gonna go right around there.

00:03:39.979 --> 00:03:43.146
So you're gonna go right around there.

00:03:44.200 --> 00:03:46.319
And I'm gonna get my ruler tool out

00:03:46.319 --> 00:03:48.973
so I can actually draw that.

00:03:48.973 --> 00:03:53.327
So I'll connect these two
dots with my ruler tool.

00:03:53.327 --> 00:03:55.614
Now remember, I'm not just
trying to graph Y equals,

00:03:55.614 --> 00:03:57.935
and in fact, this isn't
greater than or equal.

00:03:57.935 --> 00:03:59.800
It's just strictly greater than.

00:03:59.800 --> 00:04:01.824
So let me draw a dotted line there

00:04:01.824 --> 00:04:04.415
to show that I'm actually not
going to include the graph

00:04:04.415 --> 00:04:08.258
where Y equals negative two X minus 4/7.

00:04:08.258 --> 00:04:11.067
So let me do that dotted line.

00:04:11.067 --> 00:04:12.696
Dotted line right over there.

00:04:12.696 --> 00:04:16.205
Show that I'm not, it's
not going to equal to that.

00:04:16.205 --> 00:04:20.633
It's all the Y's for any X
that are greater than that.

00:04:20.633 --> 00:04:23.757
My spacing is getting a little sloppy.

00:04:23.757 --> 00:04:26.281
So that does the job.

00:04:26.281 --> 00:04:27.114
There you go.

00:04:27.114 --> 00:04:29.167
So that would be the line Y is equal

00:04:29.167 --> 00:04:33.347
to negative two X minus
4/7, but I dotted lined it

00:04:33.347 --> 00:04:35.135
because this isn't
greater than or equal to.

00:04:35.135 --> 00:04:36.986
If it was greater than or
equal to I would fill it in,

00:04:36.986 --> 00:04:39.291
but it's greater than to show
that we don't include the line

00:04:39.291 --> 00:04:41.021
but we want all of the region,

00:04:41.021 --> 00:04:43.805
all of the area above the line.

00:04:43.805 --> 00:04:45.805
So let me shade that in.

00:04:46.843 --> 00:04:49.775
So that's gonna be all of this,

00:04:49.775 --> 00:04:53.942
actually I can shade it in with
a nice, big, juicy shading,

00:04:54.993 --> 00:04:56.641
get the right tool out.

00:04:56.641 --> 00:04:57.885
Alright, here we go.

00:04:57.885 --> 00:05:02.569
So it would be all of
this area right over here,

00:05:02.569 --> 00:05:04.472
would be what I would actually shade in.

00:05:04.472 --> 00:05:05.791
So once again, I don't include the line.

00:05:05.791 --> 00:05:08.442
I include everything above the line.

00:05:08.442 --> 00:05:09.692
And I am done.


Worked example: using recursive formula for arithmetic sequence

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.902
- [Instructor] We are
told b of one is equal

00:00:02.902 --> 00:00:06.582
to negative seven, and b of n is equal

00:00:06.582 --> 00:00:11.312
to b of n minus one plus
12, and they're asking us

00:00:11.312 --> 00:00:14.645
to find the fourth term in the sequence.

00:00:16.211 --> 00:00:17.827
So what we have up here,

00:00:17.827 --> 00:00:19.910
which you could use a function definition,

00:00:19.910 --> 00:00:23.247
it's really defining
the terms of a sequence.

00:00:23.247 --> 00:00:26.336
Especially if are to input
whole numbers in here,

00:00:26.336 --> 00:00:28.749
it's the index on your sequence.

00:00:28.749 --> 00:00:30.770
What we really wanna do
is, we wanna figure out

00:00:30.770 --> 00:00:34.020
what is b of four going to be equal to?

00:00:36.376 --> 00:00:38.834
Well if we just blindly apply this,

00:00:38.834 --> 00:00:42.300
we would say all right b of four,

00:00:42.300 --> 00:00:45.981
so be of n, is equal to
b of n minus one plus 12,

00:00:45.981 --> 00:00:49.731
so it's gonna be b of
four minus one plus 12.

00:00:51.717 --> 00:00:53.342
Well four minus one is just three,

00:00:53.342 --> 00:00:57.342
so it's going to be equal
to b of three plus 12.

00:00:58.797 --> 00:01:01.031
All I did is said okay, well
we're not trying to figure out,

00:01:01.031 --> 00:01:03.331
or we're not immediately
trying to figure out

00:01:03.331 --> 00:01:04.340
what b of one is.

00:01:04.340 --> 00:01:06.471
We're trying to figure
out what b of four is,

00:01:06.471 --> 00:01:10.151
so n is equal to four, so b
of four is going to be equal

00:01:10.151 --> 00:01:14.068
to b of four minus one,
or b of three, plus 12.

00:01:15.383 --> 00:01:18.151
To evaluate this, we have to
figure out what b of three is,

00:01:18.151 --> 00:01:19.962
so let's write that down.

00:01:19.962 --> 00:01:21.808
That's what's fun about
a recursive definition.

00:01:21.808 --> 00:01:25.880
You have to keep recursing
backwards, so b of three.

00:01:25.880 --> 00:01:29.442
Well if n is three, that's
going to be equal to b of,

00:01:29.442 --> 00:01:32.859
now n minus one is two, b of two plus 12.

00:01:34.113 --> 00:01:37.393
We don't know what b of two
is, so let's keep going.

00:01:37.393 --> 00:01:39.774
We need to figure out b of two.

00:01:39.774 --> 00:01:43.530
If we use the same definition,
b of two is going to be equal

00:01:43.530 --> 00:01:47.697
to b of two minus one plus
12, so b of two minus one.

00:01:48.970 --> 00:01:53.957
That's b of one plus 12, but
we don't know what b of one is,

00:01:53.957 --> 00:01:55.841
so let's figure that out.

00:01:55.841 --> 00:01:59.557
B of one is equal to, well
here we can finally use

00:01:59.557 --> 00:02:03.724
this top clause, so b of one
is equal to negative seven.

00:02:04.741 --> 00:02:07.733
Now we can go and fill everything back in.

00:02:07.733 --> 00:02:10.514
If b of one is equal to negative
seven, then we know that

00:02:10.514 --> 00:02:13.243
this right over here is negative seven,

00:02:13.243 --> 00:02:15.956
and now we can figure out
that b of two is equal

00:02:15.956 --> 00:02:20.039
to negative seven plus 12
which is equal to five.

00:02:21.699 --> 00:02:24.598
If b of two is equal to
five, well then this is equal

00:02:24.598 --> 00:02:27.407
to five right over here,
and then now we know

00:02:27.407 --> 00:02:30.706
that b of three is equal to five plus 12

00:02:30.706 --> 00:02:32.456
which is equal to 17.

00:02:33.738 --> 00:02:37.029
If we know that b of three is equal to 17,

00:02:37.029 --> 00:02:40.948
then we're ready to calculate
what b of four is going to be.

00:02:40.948 --> 00:02:43.237
B of four is now, it's b of three,

00:02:43.237 --> 00:02:46.095
which we figured out was 17 plus 12

00:02:46.095 --> 00:02:47.845
which is equal to 29,

00:02:49.317 --> 00:02:50.650
and we are done.


Example finding appropriate units

WEBVTT
Kind: captions
Language: en

00:00:00.226 --> 00:00:03.054
- [Instructor] Luiza runs
a lawn-mowing business.

00:00:03.054 --> 00:00:05.137
She decides to measure the rate

00:00:05.137 --> 00:00:08.137
at which the volume of fuel she uses

00:00:09.816 --> 00:00:12.816
increases with the area of the lawn.

00:00:14.799 --> 00:00:19.598
What would be an appropriate
unit for Luiza's purpose?

00:00:19.598 --> 00:00:21.804
So let me reread this to
make sure I understand it.

00:00:21.804 --> 00:00:23.689
She decides to measure.

00:00:23.689 --> 00:00:26.389
She's gonna measure the rate

00:00:26.389 --> 00:00:29.389
at which the volume of fuel she uses

00:00:31.456 --> 00:00:34.456
increases with the area of the lawn.

00:00:35.292 --> 00:00:39.459
So her unit should be something
that says volume per area.

00:00:41.560 --> 00:00:44.560
She wants to see how her
volume of fuel increases

00:00:44.560 --> 00:00:46.153
with the area of the lawn.

00:00:46.153 --> 00:00:50.320
So it should be volume per
area, should be her rate.

00:00:51.630 --> 00:00:54.051
So let's think about which of these units

00:00:54.051 --> 00:00:56.051
give us volume per area.

00:00:56.921 --> 00:01:01.088
So liters right over here,
that is definitely volume

00:01:01.945 --> 00:01:04.931
but this right over here
isn't area, this is distance.

00:01:04.931 --> 00:01:07.304
If it said square kilometers
or kilometers squared

00:01:07.304 --> 00:01:08.577
then we'd be in business.

00:01:08.577 --> 00:01:10.413
So we could rule this one out.

00:01:10.413 --> 00:01:11.844
Centimeters per kilometer.

00:01:11.844 --> 00:01:15.691
Well, this is a distance per
distance, not a volume per area

00:01:15.691 --> 00:01:18.066
so we can rule that out.

00:01:18.066 --> 00:01:20.022
Centimeters per square kilometers.

00:01:20.022 --> 00:01:22.507
Well, this is a distance per area

00:01:22.507 --> 00:01:25.004
or distance per distance squared

00:01:25.004 --> 00:01:28.365
but that is not volume per area.

00:01:28.365 --> 00:01:30.454
So let's rule that one out.

00:01:30.454 --> 00:01:32.577
Now, liters per kilometer squared.

00:01:32.577 --> 00:01:34.051
This one's looking good.

00:01:34.051 --> 00:01:36.218
This is a volume in liters

00:01:37.165 --> 00:01:40.069
and this right over here is an area

00:01:40.069 --> 00:01:42.033
and this isn't the only
one that would have worked

00:01:42.033 --> 00:01:43.329
but this is out of the choices,

00:01:43.329 --> 00:01:44.914
this is the only one that works.

00:01:44.914 --> 00:01:47.572
If they gave, if it was something like

00:01:47.572 --> 00:01:51.655
a pint per foot squared
or a pint per square foot

00:01:53.061 --> 00:01:57.228
or even if it was meter
cubed per kilometers squared,

00:01:59.308 --> 00:02:02.186
that would have also been volume per area

00:02:02.186 --> 00:02:04.851
but this is the best choice of these.

00:02:04.851 --> 00:02:07.351
Let's do another one of these.

00:02:16.592 --> 00:02:19.501
Snow is piling outside Cameron's house.

00:02:19.501 --> 00:02:24.281
He decides to measure the
rate at which the height,

00:02:24.281 --> 00:02:27.864
the height of the pile
increases over time.

00:02:29.137 --> 00:02:31.967
What would be an appropriate
unit for Cameron's purposes?

00:02:31.967 --> 00:02:33.082
And like always, pause the video

00:02:33.082 --> 00:02:35.485
and see if you can figure
it out on your own.

00:02:35.485 --> 00:02:38.794
So the rate at which
height increases over time.

00:02:38.794 --> 00:02:41.729
So it should be a height per time

00:02:41.729 --> 00:02:44.224
or really I should say a distance per time

00:02:44.224 --> 00:02:46.124
is what the height is going to be doing,

00:02:46.124 --> 00:02:48.864
the rate at which the height
is changing over time.

00:02:48.864 --> 00:02:53.400
It should be a distance or
maybe even better, a length.

00:02:53.400 --> 00:02:55.150
Length per unit time.

00:02:58.109 --> 00:03:00.604
So let's see, this is hours per meter.

00:03:00.604 --> 00:03:02.980
So this is time per length.

00:03:02.980 --> 00:03:04.982
This is the reciprocal of that.

00:03:04.982 --> 00:03:09.065
This one right over here,
this is time per length

00:03:10.314 --> 00:03:14.858
as opposed to length per time
so I would rule that one out.

00:03:14.858 --> 00:03:15.839
Liters per minute.

00:03:15.839 --> 00:03:18.134
This is a volume per unit time,

00:03:18.134 --> 00:03:19.279
not a length per unit of time.

00:03:19.279 --> 00:03:20.548
Rule that one out.

00:03:20.548 --> 00:03:22.524
Minutes per liter.

00:03:22.524 --> 00:03:24.742
This is a time per volume.

00:03:24.742 --> 00:03:26.675
Well, we don't wanna do a time per volume.

00:03:26.675 --> 00:03:29.636
We want a length per time.

00:03:29.636 --> 00:03:33.020
Meters, that's a length per time, hour.

00:03:33.020 --> 00:03:33.911
That one works.

00:03:33.911 --> 00:03:35.244
Meters per hour.

00:03:36.745 --> 00:03:38.078
This is a length

00:03:39.286 --> 00:03:41.585
and that is a time.

00:03:41.585 --> 00:03:44.798
And it's good to always go
back to the original context.

00:03:44.798 --> 00:03:45.798
He decides to measure the rate

00:03:45.798 --> 00:03:48.616
at which the height of the
pile increases over time.

00:03:48.616 --> 00:03:49.449
So if someone said, hey,

00:03:49.449 --> 00:03:52.528
that height of that pile
is increasing five meters.

00:03:52.528 --> 00:03:54.275
Well, that would be too much for snow

00:03:54.275 --> 00:03:58.396
but let's say it's increasing
half a meter per hour

00:03:58.396 --> 00:04:00.240
which even that would be quite fast

00:04:00.240 --> 00:04:01.631
but half a meter per hour.

00:04:01.631 --> 00:04:02.696
That makes sense in your brain.

00:04:02.696 --> 00:04:05.644
Hey, every hour I'm gonna get
half a meter more of snow.

00:04:05.644 --> 00:04:07.732
If someone were to tell you,

00:04:07.732 --> 00:04:11.899
hey, the snow outside is
increasing at a rate of five hours,

00:04:13.661 --> 00:04:17.330
is increasing at a rate of
say five liters per minute,

00:04:17.330 --> 00:04:18.844
well, that right there,

00:04:18.844 --> 00:04:22.121
that could be maybe the volume of snow

00:04:22.121 --> 00:04:25.256
maybe over your entire lawn or something

00:04:25.256 --> 00:04:28.109
but that would not be giving
you the height per time.

00:04:28.109 --> 00:04:31.137
Height per time would
be the length per time.

00:04:31.137 --> 00:04:32.470
Right over here.


Worked examples: slope-intercept intro

WEBVTT
Kind: captions
Language: en

00:00:00.179 --> 00:00:01.799
- [Sal] Let's do some practice examples

00:00:01.799 --> 00:00:05.549
from our intro to slope
y intercept exercise.

00:00:06.422 --> 00:00:08.956
What is the slope of y is equal to

00:00:08.956 --> 00:00:12.055
negative four x minus three?

00:00:12.055 --> 00:00:13.462
So you might already recognize,

00:00:13.462 --> 00:00:15.747
this is in slope intercept form.

00:00:15.747 --> 00:00:17.842
Just as a reminder,
slope intercept form is

00:00:17.842 --> 00:00:18.925
y is equal to

00:00:19.910 --> 00:00:20.743
mx plus b,

00:00:22.510 --> 00:00:25.795
where the coefficient on
this x term right over here,

00:00:25.795 --> 00:00:27.295
that is our slope,

00:00:28.518 --> 00:00:30.362
and then this constant right over here,

00:00:30.362 --> 00:00:34.119
that is going to give
you your y intercept.

00:00:34.119 --> 00:00:36.038
So if they're saying,
what is the slope here?

00:00:36.038 --> 00:00:36.996
Well I just need to figure out,

00:00:36.996 --> 00:00:39.844
what is the coefficient on this x term?

00:00:39.844 --> 00:00:41.703
And you can see that the coefficient here

00:00:41.703 --> 00:00:43.286
is a negative four.

00:00:44.441 --> 00:00:47.054
So that is going to be our m,

00:00:47.054 --> 00:00:49.147
that is going to be our slope.

00:00:49.147 --> 00:00:50.711
Now just as a reminder,
you have to make sure

00:00:50.711 --> 00:00:52.550
that it's solved in this way,

00:00:52.550 --> 00:00:54.345
that is is solved for y,

00:00:54.345 --> 00:00:58.262
y is equal to something
times x minus three,

00:00:59.535 --> 00:01:01.129
so that's our slope.

00:01:01.129 --> 00:01:03.629
Let's do another one of these.

00:01:05.029 --> 00:01:07.919
So we're asked, what is the y intercept

00:01:07.919 --> 00:01:11.965
of y is equal to negative
three x minus two?

00:01:11.965 --> 00:01:15.684
So once again we already have
it in slope intercept form.

00:01:15.684 --> 00:01:17.562
It's already been solved for y.

00:01:17.562 --> 00:01:20.979
It's of the form y is equal to mx plus b,

00:01:22.037 --> 00:01:26.050
where m, our slope, is
given right over here,

00:01:26.050 --> 00:01:27.207
negative three.

00:01:27.207 --> 00:01:28.787
But they're not asking for our slope,

00:01:28.787 --> 00:01:31.443
they're asking for the y intercept.

00:01:31.443 --> 00:01:33.710
Well the y intercept is given by b here,

00:01:33.710 --> 00:01:35.460
so b is negative two.

00:01:36.645 --> 00:01:39.484
Pay close attention to the sign here.

00:01:39.484 --> 00:01:41.780
So b is equal to negative two.

00:01:41.780 --> 00:01:43.201
But when I look at these choices,

00:01:43.201 --> 00:01:45.132
I don't see a b is equal to negative two,

00:01:45.132 --> 00:01:47.054
so what are they talking about?

00:01:47.054 --> 00:01:48.547
Well a y intercept is,

00:01:48.547 --> 00:01:51.497
what is the y value
when x is equal to zero?

00:01:51.497 --> 00:01:52.751
And you could see that here,

00:01:52.751 --> 00:01:55.082
if x was equal to zero
then that term goes away

00:01:55.082 --> 00:01:56.723
and y is equal to b.

00:01:56.723 --> 00:01:59.191
So if you want to know the point where

00:01:59.191 --> 00:02:01.391
the graph described by this equation

00:02:01.391 --> 00:02:03.498
intercepts the y axis,

00:02:03.498 --> 00:02:08.297
well it's going to be what
is y when x is equal to zero?

00:02:08.297 --> 00:02:12.422
Well when x is equal to zero,
y is equal to negative two.

00:02:12.422 --> 00:02:15.031
And you could see that in
our original equation, again.

00:02:15.031 --> 00:02:17.097
If x were zero, this term would go away,

00:02:17.097 --> 00:02:19.562
and y would be equal to negative two.

00:02:19.562 --> 00:02:22.286
So zero, comma, negative two.

00:02:22.286 --> 00:02:24.335
So it would be that
choice right over there.

00:02:24.335 --> 00:02:26.132
On Khan Academy obviously you
just have to click on that,

00:02:26.132 --> 00:02:27.987
you don't have to shade it in.

00:02:27.987 --> 00:02:29.487
Let's do one more.

00:02:32.062 --> 00:02:35.776
Complete the equation of the
line whose slope is five,

00:02:35.776 --> 00:02:38.859
and y intercept is zero, comma, four.

00:02:39.946 --> 00:02:41.946
So once again the general form is

00:02:41.946 --> 00:02:44.613
y is equal to our slope times x,

00:02:47.065 --> 00:02:48.894
if I want to put it in
slope intercept form,

00:02:48.894 --> 00:02:50.644
plus our y intercept.

00:02:52.233 --> 00:02:55.573
Well they're telling
us, our slope is five.

00:02:55.573 --> 00:02:58.049
Whose slope is five.

00:02:58.049 --> 00:03:00.783
So we know that m is going to be five.

00:03:00.783 --> 00:03:05.198
And they tell us that the
y intercept is zero, four.

00:03:05.198 --> 00:03:08.387
So the y intercept, b,
that is the value of y

00:03:08.387 --> 00:03:10.218
when x is equal to zero.

00:03:10.218 --> 00:03:12.453
So the value of y when x equals zero

00:03:12.453 --> 00:03:14.375
is this four right over here.

00:03:14.375 --> 00:03:16.033
So that is going to be four.

00:03:16.033 --> 00:03:19.033
So I could say y is equal to

00:03:19.033 --> 00:03:21.200
five times x plus four.

00:03:25.619 --> 00:03:27.889
And when you're actually
entering it on Khan Academy,

00:03:27.889 --> 00:03:29.218
you would just type it in,

00:03:29.218 --> 00:03:30.325
or if you're using the app

00:03:30.325 --> 00:03:31.348
you would use it with your finger.

00:03:31.348 --> 00:03:33.370
And I always make the
mistake of writing y equals,

00:03:33.370 --> 00:03:35.572
and I type in y equals five x plus four.

00:03:35.572 --> 00:03:38.074
Notice they already gave you
the y equals right over there.

00:03:38.074 --> 00:03:39.242
That's all you have to do,

00:03:39.242 --> 00:03:40.776
recognize the slope, the y intercept,

00:03:40.776 --> 00:03:44.905
and then remember what the slope
intercept form actually is.


Horizontal & vertical lines

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:01.446
- [Instructor] What is the equation

00:00:01.446 --> 00:00:03.779
of the horizontal line

00:00:03.779 --> 00:00:06.932
through the point negative four comma six?

00:00:06.932 --> 00:00:08.506
So let's just visualize this.

00:00:08.506 --> 00:00:09.556
Once you get the hang of it,

00:00:09.556 --> 00:00:11.693
you might not have to draw a graph,

00:00:11.693 --> 00:00:14.407
but for explanatory
purposes, it might be useful.

00:00:14.407 --> 00:00:16.282
So negative four comma six,

00:00:16.282 --> 00:00:19.238
that's going to be in the second quadrant.

00:00:19.238 --> 00:00:21.056
So if this is my

00:00:21.056 --> 00:00:21.889
x-axis,

00:00:21.889 --> 00:00:23.111
That is my

00:00:23.111 --> 00:00:24.394
y-axis.

00:00:24.394 --> 00:00:26.999
I'm going to go negative
four in the x direction.

00:00:26.999 --> 00:00:27.940
So one,

00:00:27.940 --> 00:00:28.773
two,

00:00:28.773 --> 00:00:29.606
three,

00:00:29.606 --> 00:00:30.442
four.

00:00:30.442 --> 00:00:31.275
Negative four.

00:00:31.275 --> 00:00:32.108
And then one,

00:00:32.108 --> 00:00:32.941
two,

00:00:32.941 --> 00:00:33.774
three,

00:00:33.774 --> 00:00:34.607
four,

00:00:34.607 --> 00:00:35.440
five,

00:00:35.440 --> 00:00:36.273
six,

00:00:36.273 --> 00:00:37.640
in the y direction.

00:00:37.640 --> 00:00:38.987
So the point that we care about

00:00:38.987 --> 00:00:40.838
is going to be right over there.

00:00:40.838 --> 00:00:42.261
Negative four

00:00:42.261 --> 00:00:43.329
comma six.

00:00:43.329 --> 00:00:47.038
And what is the equation
of the horizontal line?

00:00:47.038 --> 00:00:48.925
It is a horizontal line.

00:00:48.925 --> 00:00:51.096
So it's just going to go straight

00:00:51.096 --> 00:00:53.162
left, right like this.

00:00:53.162 --> 00:00:55.450
That is what the line
would actually look like.

00:00:55.450 --> 00:00:57.557
So what is that equation?

00:00:57.557 --> 00:00:59.420
Well, for any x,

00:00:59.420 --> 00:01:02.340
y is going to be equal to six.

00:01:02.340 --> 00:01:03.733
This is the equation,

00:01:03.733 --> 00:01:05.556
y is equal to six.

00:01:05.556 --> 00:01:07.454
Doesn't matter what x you input here,

00:01:07.454 --> 00:01:08.720
you're gonna get y equals six.

00:01:08.720 --> 00:01:10.751
It just stays constant right over there.

00:01:10.751 --> 00:01:11.895
So the equation

00:01:11.895 --> 00:01:14.182
is y is equal to six.

00:01:14.182 --> 00:01:16.682
Let's do another one of these.

00:01:18.664 --> 00:01:20.887
So here we are asked

00:01:20.887 --> 00:01:23.304
what is the slope of the line

00:01:24.260 --> 00:01:26.593
y is equal to negative four?

00:01:28.608 --> 00:01:29.943
So let's visualize it

00:01:29.943 --> 00:01:30.959
and then in the future,

00:01:30.959 --> 00:01:33.559
you might not have to draw it like this.

00:01:33.559 --> 00:01:34.987
But let's just draw

00:01:34.987 --> 00:01:36.543
our axis again.

00:01:36.543 --> 00:01:38.029
X-axis

00:01:38.029 --> 00:01:39.614
y-axis

00:01:39.614 --> 00:01:41.686
and the slope of line
y equals negative four.

00:01:41.686 --> 00:01:43.677
So for whatever x you have,

00:01:43.677 --> 00:01:45.047
y is going to be negative four.

00:01:45.047 --> 00:01:47.120
Let's say that's negative
four right over there.

00:01:47.120 --> 00:01:47.953
And so,

00:01:49.035 --> 00:01:50.118
the line is y

00:01:50.963 --> 00:01:53.470
the line is y equals negative four.

00:01:53.470 --> 00:01:56.158
So I can draw it like this.

00:01:56.158 --> 00:01:57.998
So what's the slope of that?

00:01:57.998 --> 00:02:01.655
Well, slope is change in
y for given change in x.

00:02:01.655 --> 00:02:03.391
And here, no matter what I change my x,

00:02:03.391 --> 00:02:04.581
y doesn't change.

00:02:04.581 --> 00:02:06.578
It stays at negative four.

00:02:06.578 --> 00:02:09.649
My change in y over change in x.

00:02:09.649 --> 00:02:11.338
Doesn't matter what my change in x is.

00:02:11.338 --> 00:02:13.213
My change in y is always going to be zero.

00:02:13.213 --> 00:02:14.229
It's constant.

00:02:14.229 --> 00:02:18.223
So the slope here is
going to be equal to zero.

00:02:18.223 --> 00:02:20.050
Y doesn't change, no matter how much

00:02:20.050 --> 00:02:21.133
you change x.

00:02:22.788 --> 00:02:23.728
Let's do another one of these.

00:02:23.728 --> 00:02:24.728
This is fun.

00:02:27.574 --> 00:02:30.207
So now they are asking us,

00:02:30.207 --> 00:02:32.875
what is the slope of the line x

00:02:32.875 --> 00:02:35.313
equals negative three?

00:02:35.313 --> 00:02:37.235
Let me graph that one.

00:02:37.235 --> 00:02:38.663
So,

00:02:38.663 --> 00:02:41.122
I'm just going to draw my axis real fast.

00:02:41.122 --> 00:02:42.467
X-axis

00:02:42.467 --> 00:02:43.623
y-axis

00:02:43.623 --> 00:02:45.037
X is equal to negative three.

00:02:45.037 --> 00:02:46.228
So negative one,

00:02:46.228 --> 00:02:47.218
negative two,

00:02:47.218 --> 00:02:48.468
negative three.

00:02:49.391 --> 00:02:52.035
And so, this line is going to look

00:02:52.035 --> 00:02:52.868
let me,

00:02:53.884 --> 00:02:55.013
it's going to look like this.

00:02:55.013 --> 00:02:56.044
No matter what y

00:02:56.044 --> 00:02:57.179
or

00:02:57.179 --> 00:02:58.623
you can say no matter what y is.

00:02:58.623 --> 00:03:00.866
X is going to be equal to negative three.

00:03:00.866 --> 00:03:01.897
So,

00:03:01.897 --> 00:03:02.977
it would look like this.

00:03:02.977 --> 00:03:05.394
X is equal to negative three.

00:03:06.265 --> 00:03:08.828
So what's the slope here?

00:03:08.828 --> 00:03:10.215
Well,

00:03:10.215 --> 00:03:11.371
it's undefined.

00:03:11.371 --> 00:03:14.638
A vertical line has an undefined slope.

00:03:14.638 --> 00:03:16.672
Remember, you want to do
what's your change in y

00:03:16.672 --> 00:03:18.484
or change in x.

00:03:18.484 --> 00:03:20.087
Change in y

00:03:20.087 --> 00:03:21.648
or change in x.

00:03:21.648 --> 00:03:23.331
Well, you can think about what's the slope

00:03:23.331 --> 00:03:24.475
as you approach this

00:03:24.475 --> 00:03:26.293
but once again, that could be,

00:03:26.293 --> 00:03:27.126
some people would say,

00:03:27.126 --> 00:03:29.581
maybe it's infinite, maybe
it's negative infinity.

00:03:29.581 --> 00:03:31.191
But that's why it's undefined.

00:03:31.191 --> 00:03:35.684
A vertical line is going
to have an undefined slope.

00:03:35.684 --> 00:03:37.676
So we'll go with undefined.

00:03:37.676 --> 00:03:39.176
Let's do one more.

00:03:41.277 --> 00:03:43.667
What is the equation of the vertical line

00:03:43.667 --> 00:03:47.296
through negative five comma negative two?

00:03:47.296 --> 00:03:48.129
So,

00:03:48.129 --> 00:03:49.734
let me do this without even drawing it.

00:03:49.734 --> 00:03:51.538
I'll draw it right after that.

00:03:51.538 --> 00:03:53.482
So, if we're talking about a vertical.

00:03:53.482 --> 00:03:56.122
If we're talking about a vertical line,

00:03:56.122 --> 00:03:58.030
that means that x doesn't change.

00:03:58.030 --> 00:03:59.375
X

00:03:59.375 --> 00:04:00.859
doesn't

00:04:00.859 --> 00:04:01.785
change.

00:04:01.785 --> 00:04:03.311
If we were talking
about a horizontal line,

00:04:03.311 --> 00:04:05.101
then we'd say y doesn't change.

00:04:05.101 --> 00:04:07.024
So if x doesn't change,

00:04:07.024 --> 00:04:07.992
that means that x

00:04:07.992 --> 00:04:11.517
is just going to be equal
to some constant value.

00:04:11.517 --> 00:04:13.836
Well, if it contains
the points negative five

00:04:13.836 --> 00:04:15.675
comma negative two,

00:04:15.675 --> 00:04:18.151
so if it has a point where
x is equal to negative five

00:04:18.151 --> 00:04:20.958
and if x never changes,
it's a vertical line,

00:04:20.958 --> 00:04:22.305
well that means its equation has to be

00:04:22.305 --> 00:04:24.638
x is equal to negative five.

00:04:25.850 --> 00:04:27.787
And we can draw that out,

00:04:27.787 --> 00:04:28.787
if it helps.

00:04:29.786 --> 00:04:31.841
So let me draw that out.

00:04:31.841 --> 00:04:32.963
So,

00:04:32.963 --> 00:04:35.979
I need to make sure
that's a straight line.

00:04:35.979 --> 00:04:36.812
Okay,

00:04:36.812 --> 00:04:37.971
so we have x,

00:04:37.971 --> 00:04:39.138
and we have y,

00:04:40.221 --> 00:04:42.983
so we have the point negative
five comma negative two.

00:04:42.983 --> 00:04:45.115
So negative one,

00:04:45.115 --> 00:04:45.958
two,

00:04:45.958 --> 00:04:46.791
three,

00:04:46.791 --> 00:04:47.624
four,

00:04:47.624 --> 00:04:48.995
five,

00:04:48.995 --> 00:04:50.102
negative one,

00:04:50.102 --> 00:04:51.085
two

00:04:51.085 --> 00:04:52.756
so we want to go have a vertical line

00:04:52.756 --> 00:04:54.428
that goes through that point.

00:04:54.428 --> 00:04:55.261
So a vertical line,

00:04:55.261 --> 00:04:57.273
well that just goes straight up and down.

00:04:57.273 --> 00:04:59.721
So it's just going to look like this.

00:04:59.721 --> 00:05:00.784
And so notice,

00:05:00.784 --> 00:05:02.462
x never changes.

00:05:02.462 --> 00:05:03.918
No matter what y is,

00:05:03.918 --> 00:05:06.983
x is equal to negative five.

00:05:06.983 --> 00:05:08.418
This has an undefined slope.

00:05:08.418 --> 00:05:09.401
It is a vertical line.

00:05:09.401 --> 00:05:13.318
Its equation is x is
equal to negative five.