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## Algebra 1 (Eureka Math/EngageNY)

### Unit 4: Lesson 14

Topic C: Lessons 20-22: Scaling and transforming graphs- Scaling & reflecting parabolas
- Scale & reflect parabolas
- Identifying function transformations
- Identifying horizontal squash from graph
- Reflecting & compressing functions
- Identify function transformations
- Transforming the square-root function
- Graphs of square-root functions
- Square-root functions & their graphs
- Radical functions & their graphs
- Graphs of square and cube root functions

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# Scaling & reflecting parabolas

CCSS.Math:

The graph of y=k⋅x² is the graph of y=x² scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of a parabola from its graph.

## Want to join the conversation?

- Where/How did he get 1/4? Why isn't the work for THAT shown?(27 votes)
- It helps me to compare it to the function y = -x^2, so when x = 1 or -1, y = 1, you have points (1,-1)(-1,-1). So when you widen this parabola, you need some fraction in front. When x = 2, you get x^2 = 4, so what do you fraction do you need to have this give a y value of -1? You have to multiply by the negative reciprocal, and that is where the -1/4 comes from, f(x) = - 1/4 x^2, thus f(2) = -1/4 (2)^2 = -1. So if you moved it over one more to get to x = 3, the fraction would have to be -1/9, etc.(11 votes)

- Does y2/y1 gives the scale value? For example, in this video, y1 (when x = 1) = 1 and y2 = -1/4, so -1/4/1 gives -1/4.(13 votes)
- Yes you are absolutely correct. The scale value is essentially the ratio between the the y-value of the scaled parabola to the y-value of the original parabola at a given x-value.(7 votes)

- How can you solve the problem if you don't have the graph to help you?(15 votes)
- in what situation? What kind of problem would you have like this¿(0 votes)

- The parabola y=x^2

is scaled vertically by a factor of 7.

What is the equation of the new parabola?

what would be the answer for this?(5 votes) - How do you find the stretch/shrink factor? As in, how did he get 1/4?(3 votes)
- For the parent function, y=x^2, the normal movement from the origin (0,0) is over 1 (both left and right) up one, over 2 (both left and right) up 4, over 3 (both ...) up 9 based on perfect squares. So your scale factor compares to that, in this case, over 2 goes down 1, so it is 1/4 that of the parent function. The same is true at 4 which is down 4 (which is 1/4 of the parent function which would be at 16 (4^2=16). So the scale factor is a change from the parent function.(4 votes)

- How can you solve the problem if you don't have the graph to help you?(2 votes)
- Start from a parent quadratic function y = x^2. Adding parameters to this function shows both scaling, reflecting, and translating this function from the original without graphing. So you may see a form such as y=a(bx-c)^2 + d. The parabola is translated (c,d) units, b reflects across y, but this just reflects it across the axis of symmetry, so it would look the same. A negative a reflects it, and if 0<a<1, it vertically compresses the parabola, and if x>1, it vertically stretches the parabola. We can do a lot with equations.(2 votes)

- How would reflecting across the y axis differ? I'm so confused.(1 vote)
- this really doesnt help at all, im still just as confused, just about different things now(2 votes)
- comparing between g(x) and y = -x^2, the y value is -1 as opposed to -4, and -1 is 1/4 of -4 so that's the scale. you can basically just take g(1) divided by f(1) (-1 divided by 4) and it'll be the scale (-1/4).(1 vote)

- How did he get -1/4?2:20-2:50(2 votes)
- That is the question with the most votes, it has been already answered.(1 vote)

## Video transcript

- [Instructor] Function
G can be thought of as a scaled version of F
of X is equal to X squared. Write the equation for G of X. So like always, pause this video and see if you can do it on your own. Alright now, let's work
through this together. So the first thing that
we might appreciate is that G seems not only to
be flipped over the x-axis, but then flipped over
and then stretched wider. So let's do these in steps. So first let's flip over, flip over the x-axis. So if we were to do this
visually it would look like this. Instead when X is equal to zero, Y is still gonna be equal to zero. But when X is equal to negative one, instead of Y being equal to one, it'd now be equal to negative one. When X is equal to one,
instead of squaring one and getting one, you then
take the negative of that to get to negative one. So when you flip it, it looks like this. Y when is X is equal to negative two instead of Y being equal to four, it would now be equal to negative four. So it would look like this. So as we just talk through
as we're trying to draw this flipped over version, whatever Y value we were
getting before for a given X, we would now get the opposite
of it, or the negative of it. So this green function right over here is going to be Y is equal
to the negative of F of X, or we could say Y is equal
to negative X squared. Whatever the X is, you square it, and then you take the negative of it. Whatever X is, you square it, and then you take the negative of it, and you see that that will
flip it over the x-axis. But that by itself does
not get us to G of X. G of X also seems to be stretched in the horizontal direction. And so let's think about,
can we multiply this times some scaling factor so
that it does that stretching so that we can match up to G of X? And the best way to do
this is to pick a point that we know sits on G of X,
and they in fact give us one. They show us right over
here that at the point two comma negative one, sits on G of X. When X is equal to two, Y is equal to negative one on G of X. So you could say G of two is negative one. Now on our green function,
when X is equal to two Y is equal to negative four. So let's see. Maybe we can just multiply
this by 1/4 to get our G. So let's see. If we were to, let's
see if we scale by 1/4, does that do the trick? Scale by 1/4. So in that case, we're gonna have Y is equal to not just negative X squared, but negative 1/4 X squared. And if you're saying hey,
so how did you get 1/4? Well I looked at when X is equal to two. On our green function,
when X is equal to two I get to negative four. Well we want that when X is equal to two to be equal to negative one. Well negative one is 1/4 of negative four, so that's why I said
okay, well let's up take to see if we could take
our green function, and if I multiply it by 1/4, that seems like it will
match up with G of X. And so let's verify that. When X is equal to
zero, well this is still all gonna be equal to
zero so that makes sense. When X is equal to one, let me do this in another color, when X is equal to one, then one squared times negative 1/4, well that does indeed look
like negative 1/4 right there. When X is equal to two,
two squared is four, times negative 1/4 is indeed
equal to negative one. Let's try this point
here 'cause it looks like this is sitting on our graph as well. When X is equal to four,
four squared is 16. 16 times negative 1/4 is
indeed equal to negative four. And it does work also for the
negative values of X as well. So I'm feeling really good that this is the equation of G of X. G of X is equal to negative
1/4 times X squared. And so in general, that
when we were saying we were scaling it, we're
scaling it by negative value. This is what flips it over the x-axis, and then multiplying it by this fraction that has an absolute value less than one, this is actually stretching it wider. If this value right over here, its absolute value was greater than one, then it would stretch it vertically, or would make it thinner in
the horizontal direction.