Sec. 2.5 Transformations of Functions

Vertical and Horizontal shifts Transformations Vertical and Horizontal shifts

Vertical and Horizontal shifts h(x)=f(x)+c shifts graph up c units h(x)=f(x)-c shifts graph down c units h(x)=f(x-c) shifts graph to the right c units h(x)=f(x+c) shifts graph to the left c units

Graph with your calculator f(x)=x2 This is a family of functions. Notice the graph is the same size and shape but at a different location. f(x)=x2+2 f(x)=x2-2 f(x)=(x-2)2 f(x)=(x+2)2

Mirror or flip the image Reflections Mirror or flip the image

Reflection in the coordinate axes If y=f(x) h(x)=-f(x) reflects in x-axis (flips over x-axis) h(x)=f(-x) reflects in y-axis (flips over y-axis)

Use your calculator to graph f(x)=√x Graph g (x) = -√x It flips over the x axis Graph h(x) = √-x It flips over the y axis

Two types of transformations Rigid- shape of graph is unchanged (Position changed) Non-Rigid-distortion- changes shape of original graph

Vertical Stretch &Vertical Shrink Caused by multiplying a function by a number

If c>1 in g(x)=c•f(x) then it is a stretch If 0<c<1 (fraction) in g(x)=c•f(x) then it is a shrink

Graph f(x)=|x| Graph h(x)=3|x| This is vertical stretch Graph g(x)=1/3|x| This is vertical shrink

Horizontal stretch and shrink A nonrigid transformation of y = f(x) represented by h(x) = f(cx) where the transformation is a horizontal shrink if c>1 and a horizontal stretch if 0<c<1. Ex 5)