Horizontal translation for the parabola is changed by the value of a variable, h, that is subtracted from x before the squaring operation. So, our starting or reference parabola formula looks like this:

**y = x ^{2}**

And our equation that includes a ** horizontal translation** looks like this:

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

So, if ** h = 6**, we say that the reference parabola is

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

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

What follows is an animation that presents many
horizontal translations for our reference parabola. Note that the value for
** h** is subtracted from

Please understand that ** x^2** means

It may not seem correct at first that you
must subtract from x to move the parabola to the right, but that is
how it works. Subtractions on the x-axis are usually related with
movements to the left, but not here. If you want to move the parabola to the
right, say, ** 4** units, then you must subtract

Be sure to understand that the subtraction
from x occurs before the squaring. Imagine that we translate the
reference parabola ** 3** units to the

y = (x - h)^{2}

y = (x - 3)^{2}

y = (5 - 3)^{2}

y = (2)^{2}

y = 4

How do you translate the reference parabola
to the left? Well, you subtract a ** negative**
number from

y = (x - h)^{2}

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

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

y = (2 + 6)^{2}

y = (8)^{2}

y = 64

So, if you wish to move the reference parabola to the right, subtract a positive number from x.

If you want to move the reference parabola to the left, subtract a negative number from x. Since we are subtracting a negative number, this could look like an addition. Here's an example:

** Subtracting** a

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

Can look like an ** addition**:

y = (x + 3)^{2}

So, a graph of this function:

y = (x + 4)^{2}

Which we should think of as:

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

Would look like the reference parabola shifted to the *left*** 4**
units:

And a graph of this function:

y = (x - 5)^{2}

Would look like the reference parabola slid to the *right*** 5** units:

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

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