Vertical scaling for the parabola is changed by the coefficient of x^{2}.
The variable ** a** is often used for this coefficient.
So, to begin, our starting or reference parabola formula looks like this:

**y = x ^{2}**

Since ** a** is the coefficient of

**y = ax ^{2}**

So, if ** a = 3**, we say that the reference parabola is

**y = 3x ^{2}**

Here's the graph for this scaling. The ** reference** parabola (

What follows is an animation that presents many
vertical scalings for our reference parabola. Note that the value for
** a** is shown within parentheses before

Please understand that ** x^2** means

Let's cover some notation. First a bit
about ** parentheses**. This:

(3)x^{2}

Is the same as:

3x^{2}

And this:

(-3)x^{2}

Is the same as:

-3x^{2}

Now, what about when a = 1 or -1? Note that this:

y = (1)x^{2}

Is simply:

y = x^{2}

And note that:

y = (-1)x^{2}

Is:

y = -x^{2}

In all of the cases shown above understand that the calculation for y proceeds this way:

- First, get the value for
**x**and square it. - Second, multiply the value for
**x**by^{2}, the vertical scaling factor.*a*

So, if x = 4 and a = -2, then:

y = ax^{2}

y = -2x^{2}

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

y = (-2)(16)

y = -32

Don't do any thinking like the following. Do
not multiply ** a** times

y = ax^{2}

y = -2x^{2}

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

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

y = (-8)^{2}

y = 64 (Not correct.)

Square
**x** first, and then multiply by ** a**.

And note that if, say, x = 3 and a = -1, then:

y = ax^{2}

y = (-1)(3^{2}) = -3^{2}

y = -(3^{2})

y = -9

About reflections: Let's for the
moment consider two cases. We will let ** a = 1** and

Now, let ** x = 2**,
and our calculation for

y = ax^{2} = (1)(2^{2}) = 4

Here's the calculation when ** a = -1**:

y = ax^{2} = (-1)(2^{2}) = -4

Here's what this all looks like:

Notice that the first point, created when ** x = 2**
and

The second point, created when ** x = 2**
and

The first point is just as far above the x-axis as the
second point is below it. We say the the second point is the **
reflection** of the first point

A similar reflection across the x-axis would occur for any x-coordinate. For example, if ** x = 3**, then

a = 1 y = ax ^{2}y = (1)(3 ^{2})y = 9 |
a = -1 y = ax ^{2}y = (-1)(3 ^{2})y = -9 |

Below we will see a graph showing how this all looks when full parabolas
are drawn. Realize that when ** a = 1** we have our
reference parabola:

y = (1)x^{2} = x^{2}

When ** a = -1**, we have:

y = (-1)x^{2} = -x^{2}

When ** a = -1**, all the points on the reference
parabola have been

Understand that the ** sign** of

So, if we know the graph of ** y = x^{2}**, we can
understand that the graph of

Now, let's explain the term 'vertical scaling factor'.
For a short time set aside the idea of reflection across the x-axis and
work only with positive values of ** a**. Imagine our
reference parabola:

If we say that it will be vertically scaled by a factor of 2, what we mean is that every point on the reference parabola has been moved up twice as high (factor of 2) from the x-axis to become the transformed parabola. So, if we have our reference parabola:

y = x^{2}

And we locate at ** x = 3**, then

y = x^{2} = 3^{2} = 9

And our transformed parabola, vertically scaled by a factor of 2, is:

y = 2x^{2}

And we again let ** x = 3**, then

y = 2x^{2} = (2)(3^{2}) = (2)(9) =
18

Notice that at the same x-coordinate, ** x = 3**,
the y-coordinate is twice as high,

So, we say that we have ** vertically scaled** the

Below are the two parabolas. The
** reference** parabola is colored

Notice that for corresponding x-coordinates, the ** y-coordinates** on the

In the above graphs note how the reference parabola's ** y-coordinates** are

If the vertically scaling factor is less than 1, then the y-coordinates
on the transformed parabola are ** less than** the corresponding y-coordinates on
the reference parabola. The transformed parabola is a squashed version of
the reference parabola in this case.

Specifically, at ** x = 3**, the reference
y-coordinate is

y = x^{2} = 3^{2} = 9

Let's let ** a = 0.5**, and we see that at

y = 0.5x^{2} = (0.5)(3^{2}) =
(0.5)(9) = 4.5

This transformed y-coordinate is exactly half (0.5) the value of the reference y-coordinate:

(0.5)(9) = 4.5

Here's the reference and transformed parabolas (colored as before) showing other examples of halving of the y-coordinates at the x-locations used on graphs earlier:

So, the vertical scaling factor controls the stretching or the compressing of the y-coordinates of the reference parabola. After this stretching or compressing of the reference parabola, we are left with a graph of the transformed, vertically scaled parabola.

Let's put x-axis reflections and vertical
scalings together. They are both controled by the value of **
a**, the coefficient of

- If
is**a***positive*), then the reference parabola*a > 0*. You might say the transformed parabola 'opens up'.*is not reflected over the x-axis* - if
is**a****negative**), then the reference parabola*a < 0*. It 'opens down'.*is reflected across the x-axis* - The reference parabola is stretched or compressed by the
*size*, of**absolute value**.*a*- If the absolute value of
is*a*than*greater*, the reference parabola is vertically*1*. All of the referenced y-coordinates become greater due to multiplication by a number greater than 1.*stretched* - If the absolute value of
is*a**less*. All of the referenced y-coordinates become less due to the multiplication by a number less than 1.*compressed*

- If the absolute value of

It really doesn't matter if you imagine the possible x-axis reflection first and then the vertical scaling, or if you imagine the scaling and then the possible reflection.

However, as we shall see later in this section about function transforms, often the order in which you view the features of the transform can be important, especially when translations (right, left, up, or down movements) are present.

If ** a = -4**, then we would have:

So, a graph of this function:

y = 4x^{2}

Would look like the reference parabola after it did not reflect across the x-axis, and after all of the y-coordinates of the reference parabola were multiplied by 4.

And a graph of this function:

y = -0.3x^{2}

Would look like the reference parabola after all of its y-coordinates were multiplied by 0.3 and then reflected across the x-axis.

Here is an EZ Graph example of this vertical scaling. Press the 'Draw graph' button.

You can change the value for ** a** using the upper
left input boxes. Press the 'Draw graph' button after you change