is the graph of function f right over here.

And then they tell us that function g is defined as

g of x is equal to one third f of x.

What is the graph of g?

And if we were doing this on Khan Academy,

this is a screenshot from our mobile app,

it has multiple choices,

but I thought we could just try to sketch it.

So pause this video, maybe in your mind,

imagine what you think the graph of g is going to look like,

or at least how you would tackle it.

All right, so g of x is equal to one third f of x.

So, for example, we can see here that f of three

is equal to negative three.

So, g of three should be one third that,

so it should be negative one.

Likewise, so, g of three would be right over there,

and likewise, g of negative three, what would that be?

Well, f of negative three is three,

so g of negative three is going to be one third that,

or it's going to be equal to one.

F of zero is zero, one third of that is still zero,

so g of zero is still going to be right over there.

And we know that's going to happen there and there as well,

and so we already have a sense of what

this graph is going to look like.

The function g is going to look something like,

something like this.

I'm just connecting the dots

and they did give us some dots

that we can use as reference points,

so the graph of g is going to look something like this.

It gets a little bit flattened out

or a little bit squooshed or smooshed a little bit

to look something like that

and you would pick the choice that looks like that.

Let's do another example.

So, here we are told this the graph of f of x

and it's defined by this expression.

What is the graph of g of x,

and g of x is this.

So pause this video and think about it again.

All right, now the key realization is,

is it looks like g of x is,

if you were to take all the terms of f of x

and multiply it by two,

or at least if you were to multiply

the absolute value by two,

and then if you were to multiply this negative two by two.

So it looks like g of x is equal to two times,

two times f of x.

And we could even set up a little table here,

this is another of the way that we can think about it.

We can think about x, we can think about f of x,

and now we can think about g of x,

which should be two times that.

So we can see that when x is equal to zero,

f of x is equal to one,

so g of x should be equal to two

because it's two times f of x.

So g of x is going to be equal to--

Or g of zero, I should say, is going to be equal to two.

What about when at x equals,

we'll say when x equals three.

When x equals three, f of x is negative two.

G of x is going to be two times that,

'cause it's two times f of x,

so it's going to be negative four.

So, g of x, or I should say g of three

is going to be negative four.

It's going to be right over there.

And then maybe let's think about one more point.

So, f of five is equal to zero.

G of five is going to be two times that,

which is still going to be equal to zero,

so it's going to be right over there.

And so the graph is going to look something like this,

I'm just really just connecting,

I'm just connecting the dots,

trying to draw some straight lines.

It's going to look something like this,

you can see it's kind of stretched

in the vertical direction.

So, if you were doing this on Khan Academy,

it'd be multiple choice,

you'd look for the graph that looks like that.

Let's do a few more examples.

So, here we're given a function g

is a vertically scaled version of f.

So we can see that g is a vertically scaled version of f.

The functions are graphed where f is a solid

and g is dashed.

Yeah, we see that.

What is the equation of g in terms of f?

So, pause this video and try to think about it.

Well, the way that I would tackle this is once again,

let's do it with a table

and let's see the relationship between f and g.

So, this column is x,

this column is f of x,

and then this column is g of x.

I'll make another column right over here.

And so, let's see some interesting points.

So, when, actually, I could pick zero,

but zero is maybe less interesting

than this point over here.

So, this is when x is equal to negative three.

F of negative three is negative three.

What is g of negative three?

It looks like it is negative nine.

When f is, when x is zero,

f of zero is negative two.

What is g of zero?

It is equal to two negative six.

And so we already see a pattern forming.

Whatever f is, g is three times that.

Whatever f is, g is three times that.

And so we don't even actually need these big columns,

but we can see that g of x is equal to three times f of x.

So that is the equation of g in terms of f.

Let's do one more example.

So, once again, they give us f of x,

this time, they're telling us the expression for f of x,

and they're telling g is a vertically scaled version of f.

The functions are graphed where f is solid and g is dashed

just like before.

What is the equation of g?

So, pause the video again.

Try to work on it.

All right, well, I'll tackle it the same way

that we did the last one.

I'm going to make a table,

so x, and then I'm going to have another column for f of x,

and then I'm going to have another column for g of x.

Now let's pick some interesting values.

So, when x is equal to one,

f of one is equal to two,

g of one is equal to eight.

Interesting.

All right, let's pick another value, let's see,

when x is equal to four,

g, or I should say f of four is equal to negative one.

When x is equal to four, f of four is equal to negative one,

yeah, I got that right.

What is g of four?

It's equal to negative four.

So it looks like, and I could try it with other points,

f of zero, when x is zero, f of zero's zero,

g is zero as well.

And so, it's clear that from these points

that g of x is four times f of x,

in all of these cases, to go from f of x to g of x,

I multiply by four.

I am multiplying by four,

zero times four is still zero.

So we could write that g of x is equal to four times f of x.

But we aren't done.

They're asking what is the equation of g.

And I think on Khan Academy, if you do this,

they might give some multiple choice,

or you actually you might be able to type it in,

but either way, I think they want the expression

in terms of an actual algebraic expression,

not just in terms of f of x.

So we could rewrite this as

g of x is equal to four times

what is f of x?

It's all of this business.

Negative six log base two of x plus eight,

and so we distribute that for g of x

is equality to four times negative six

is negative 24 log base two of x

plus four times eight is 32.

And we are done.