What happens if we change “h” in f(x) = a(x – h)² + k?

The parameter “h” translates (moves) the graph horizontally.

• If “h” is positive (h > 0), the parabola moves to the right.
• If “h” is negative (h < 0), the parabola moves to the left.

Example

Describe the effect on the graph of y = (x − 1)² if the equation is changed to y = (x + 2)² ?

Solution

If y = (x – 1)² ⇒ h = 1.

This is the parent graph translated one unit to the right, since h = 1 and 1 > 0.

If y = (x + 2)² = (x − -2) ⇒ h = -2.

This is the parent graph translated two units to the left, since h = -2 and -2 < 0.

Comparing the graphs shows that y = (x − 1)² is 3 units to the right of y = (x + 2)².

Example

If the graph of f(x) = 2(x – 3)² + 1 is to be translated 4 units to the left, what is the equation for the new parabola?

Solution

A translation 4 units to the left is horizontal and affects ONLY the “h” in the equation.

If f(x) = 2(x – 3)² + 1 ⇒ h = 3.

To translate this graph to the left, you should subtract from “h”.

Starting at 3 and moving to the left 4 units gives you 3 – 4 = -1, so the new “h” is -1.

New equation: f(x) = 2(x – -1)² + 1 which simplifies to f(x) = 2(x + 1)² + 1.

Let's see this in action.