Vertical translation for the parabola is changed by the value of a variable, k, that is added into the calculation for y after x is squared. So, our starting or reference parabola formula looks like this:

**y = x ^{2}**

And our equation that includes a vertical translation looks like this:

**y = x ^{2} + k**

So, if ** k = 5**, we say that the reference parabola is

**y = x ^{2} + 5**

Here's the graph for this translation. The reference parabola ( *y = x*^{2} )
is drawn in transparent light gray, and the transformed parabola which is
vertically translated 5 units ( *y = x*^{2}** + 5** ) is drawn in
black:

What follows is an animation that presents many
vertical translations for our reference parabola. Note that the value for
** k** is added into the calculation for

Please understand that ** x^2** means

It is easy to see how this vertical
translation moves the reference parabola up or down. Let us do a
simple calculation that will get us from the ** reference** parabola to the

y = x^{2}

y = 3^{2}

y = 9

Now, notice the formula for the ** transformed** parabola with a

y = x^{2} + k

Since the squaring of ** x** occurs before the
addition of

Here's the calculation for the y-coordinate on the ** transformed**,

y = x^{2} + k

y = 3^{2} + k

y = 3^{2} + 5

y = 9 + 5

y = 14

Hopefully, it is clear that on this ** transformed**,

To transform the reference parabola down, we
make k negative. This can look like a subtraction. Let's work at
** x = 5** and perform a

y = x^{2} + k

y = 5^{2} + k

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

y = 5^{2} - 3

y = 25 - 3

y = 22

Let's look at the graph of this function:

y = x^{2} - 6

Which we think of as:

y = x^{2} + (-6)

This should look like the ** reference** parabola translated

This function:

y = x^{2} + 4

Will look like the ** reference** parabola translated

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

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