Vertical scaling and reflection along with horizontal and vertical translations for the parabola can all happen in one equation using a, h. and k. So, our starting or reference parabola formula looks like this:

**y = x ^{2}**

And our equation that includes the scalings, reflections and translations looks like this:

**y = a(x - h) ^{2} + k**

So, if ** a = -0.1** and

**y = (-0.1)(x - (-4)) ^{2} + (-5)**

Or, with a bit of simplification:

**y = -0.1(x + 4) ^{2} - 5**

Here's the graph for this ** transform**. The

What follows is an animation that presents many x-axis reflections, and vertical scalings, and horizontal and vertical translations for our reference parabola.

Please understand that ** x^2** means

We will take a look at a y-coordinate
calculation for the transformed parabola. Let's assume that the
** reference** parabola is

y = a(x - h)^{2} + k

y = a(5 - h)^{2} + k

y = -4(5 - (-3))^{2} + (-2)

y = -4(5 + 3)^{2} - 2

y = -4(8^{2}) - 2

y = (-4)(64) - 2

y = -256 - 2

y = -258

Using the above transform ( a = -4, h = -3,
and k = -2 ), consider the following question. If a point on the
** reference** parabola is (5, 25), then what are the x- and y-coordinates on the
corresponding point on the

About the transformed y-coordinate: Let's see, at the ** reference** point (5, 25) the y-coordinate is

We still have the value of ** k** to use for the
calculation of the

Basically, we have multiplied the ** reference**
y-coordinate by

y_{transformed} = (a)(y_{reference})
+ k

y_{transformed} = (-4)(25) + (-2)

y_{transformed} = (-100) + (-2)

y_{transformed} = -102

About the transformed x-coordinate: Along the x-axis we have *no***
reflection** or

To get the ** transformed** x-coordinate in this case we simply add

x_{transformed} = x_{reference} + h

x_{transformed} = (5) + (-3)

x_{transformed} = 2

We can say now that point (5, 25) on the ** reference** parabola is

Reflect and scale first, then translate.
In the above discussion we imagined a point on the ** reference**
parabola (5, 25) being moved to a corresponding point on the

You can do the y-thinking first, or you can do the x-thinking first. The x- and y-thinking operate independently.

We'll considered the y-coordinate first. Note that we thought about vertical
reflection and vertical scaling ** before** we
considered vertical translations. You should work this way in your thinking.

- Reflect and scale
- Translate

When you reflect and scale, you can reflect first and then scale, or scale first and then reflect. You'll get to the same value. Just be sure to get done with your reflecting and scaling thinking before you go further with the translation.

And although this current discussion is about a y-coordinate, the same could be said about the x-coordinate. No matter which axis you are working with, reflect and scale first and then translate.

Now, let's look at the x-coordinate. In our parabola discussion the x-coordinate was not examined regarding y-axis reflections and horizontal scalings. It was only horizontally translated. So, calculations regarding reflections and scaling do not come up. If things were different, and they did, then we would reflect and scale first before translating along this x-axis.

We are only translating here, its the only thing to do.

The reasons why we do not consider y-axis reflections and horizontal scaling for the parabola will be covered in detail in another part of this Function Transforms section. Basically, for the moment, the parabola is symmetrical about the y-axis, so reflecting it over the y-axis does not really make an new shape.

Also, the parabola is a shape where a vertical stretch can yield the same shape as a horizontal compression. In other words for every horizontal scaling there is an equivalent vertical scaling that leads to the same transformed shape. So, all parabola scaling transforms can be imagined using only vertical thinking, as we have done here.

Again, we could
have figured out that ** x = 2** on the transformed
parabola before we found that

Here is an EZ Graph example of these scalings and translations:

You can change the value for ** a**,