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## Algebra 2

# Scaling functions vertically: examples

CCSS.Math:

The function k⋅f(x) is a vertical scaling of f. See multiple examples of how we relate the two functions and their graphs, and determine the value of k.

## Want to join the conversation?

- It looks like a parabola to me, why isn't it f(x^2)?(5 votes)
- f(x) just represents the output of the function, not the contents of the function itself. Since it is a parabola, it will probably have a kx^2 term in it (where k is a constant), but the output would still be f(x).(6 votes)

- 1:06Why do we know the points (5,0) and (-5,0)? Shouldn't you apply the same logic to these points as to the first examples, and multiply the f(x) by 1/3?

If this question doesn't make sense, I'm not surprised. I suck at math.

HEEEEEEEEEEEEEEELP!(4 votes)- Exactly on point !

We know the point (-5,0) and (5,0) because it is already defined in the graph. The original function passes through those points, and that's why we know them.

In addition, we can multiply f(x) by 1/3. Just remember f(x) concerns with only the output**not**the input.

^^ Just to clarify that I mean, refer to the same problem at1:06

f(-5) = 0

0*1/3 = 0

f(-2) = 2

2 * 1/3 =**2/3**

f(3) = -3

-3 * 1/3 = -1

f(5) = 0

0*1/3 = 0

hopefully this helps !(4 votes)

- In the 1st graph, since f(x) is multiplied by 1/3, shouldn't the y-intercept change as well?(3 votes)
- Think about it like this:

It looks like the y-intercept for f(x) is at y=0.

If the whole equation is being scaled by 1/3, then to find out what the y-intercept will be for g(x), we should multiply the y-intercept from f(x) by 1/3 to find the new y-intercept.

1/3*0=0, so the y-intercept stays the same for this graph.

Keep going!(3 votes)

- the parabola y = x^2 is scaled vertically by a factor of 7. What is the equation of the new parabola.(3 votes)
- Answer: y= 7x^2

You can see that for every value for x^2, the y coordinate becomes seven times larger.

x=1; y=1 After scaling x=1; y=7

x=2; y=4 '""" x=2; y=28

etc.

General formula y=cx^2, whereby c is the scaling factor (vertically)(1 vote)

- When he makes the second graph, doesn't he forget to translate it downward?(2 votes)
- He does not, I would recommend checking the question again(2 votes)

- Couldn't we assume that since it is an odd function and the original function hits the points at (-3, 3) and (3, -3) that it would be the same but 1/3 of the way instead of individually finding where 1/3rd of the point would be for each point unless finding if it's odd or even has no pertinence or it can't always by applied this way??(2 votes)
- Question: Why in the second graph do we multiply the whole function by 2 and not only the absolute value x-3?(1 vote)
- When you have a Problem in the form -

A*|x - B| + C

Then -

A = scale factor or multiplier (you need to find A*|x-B| so multiply |x-B| by A)

B = horizontal shift (if B is positive shift to the NEGATIVE direction and vice versa)

C = vertical shift (If C is positive shift to the positive direction and vice versa)

Note: This works for other kinds of functions as well, not just absolute value functions. For example, you could have also had a square root instead of the absolute value sign.

Hope this answered your question!(1 vote)

- Don't you have to do the opposite of what it says to do to the x?(0 votes)
- You are correct however, that applies to shifting the function(translating), here, Sal is scaling the function.(2 votes)

## Video transcript

- [Instructor] So, we're told that this 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.