Transformations of Functions 2.5 JMerrill, 2007 Contributions by DDillon Revised 2008

Common (Parent) Functions

f(x) = xIdentity

f(x) = x 3 cubic

f(x) = |x|absolute

f(x) = x 2 quadratic

Square root

f(x) = cConstant

Vertical and Horizontal Shifts

f(x) + 2 Shift up 2 units Add two to all y-values

f(x) – 2 Shift down 2 units Subtract two from y-values

f(x + 2) Shift to the left 2 units Subtract two from x-values

f(x – 2) Shift to the right 2 units Add two to all x-values

Reflections

-f(x) Reflect across the x- axis Change signs of all y-values

f(-x) Change signs of all x-values Reflect across the y- axis

Nonrigid Transformations  Stretch/Shrink  Vertical a f(x)

2f(x) Vertically stretch by factor of 2 Multiply y- values by 2

½ f(x) Vertically shrink by factor of 2 Multiply y- values by ½

Example UUse the graph of f(x) = x 2 to explain the transformations to g(x) = 2(x + 3) 2 – 1 LLeft 3 VVertical stretch by 2 DDown 1

Multiple Transformations  A function involving more than one transformation can be graphed by performing transformations in the following order: Horizontal Shifting Stretching/Shrinking Reflecting Vertical Shifting

What happened? Shift right 5 Shift up 3No stretch f(x) = -(x – 5) Reflect over x-axis