Transformations to Parent Functions

Translation (Shift) A vertical translation is made on a function by adding or subtracting a number to the function. Example: y = x + 3 (translation up) Example: y = x² - 5 (translation down) A translation up is also called a vertical shift up. A translation down is also called a vertical shift down.

Example: y = |x| + 2 Parent function (y = |x|) shown on graph in red. The transformation of the parent function is shown in blue. It is a shift up (or vertical translation up) of 2 units.)

Example: y = x - 1 Parent function (y = x) shown on graph in red. The transformation of the parent function is shown in blue. It is a shift down (or vertical translation down) of 1 unit.

Reflection A reflection on the x-axis is made on a function by multiplying the parent function by a negative. Multiplying by a negative “flips” the graph of the function over the x-axis. Example: y = -x² is a reflection of the parent function y = x².

Example: y = - x² The reflection of the parent function is shown in blue. It is a reflection over the x-axis of the function y = x²

Stretch A stretch is made on a parent function by multiplying the parent function by a number x such that |x| > 1. Example: y = 3x² A stretch is also referred to as a narrowing of the graph of the function closer to the y-axis.

Example: y = 2x The blue line shows the graph of the stretch of the parent function, y = 2x, by a scale factor of 2.

Shrink A shrink is made on a function by multiplying the parent function by a number x such that 0 < |x| < 1. Example: y = (½) x. A shrink is also referred to as a widening of the graph of the function closer to the x-axis.

Example: y = (¼) x The blue line shows the graph of the shrink of the parent function, y = (¼)x, by a scale factor of ¼.

What happens when we combine the two (blue is parent function)?