## 1-3: New functions from old functions

Vertical and Horizontal Shifts: suppose  $c> 0$  . To obtain the graph of

• $y=f(x)+c$  shift the graph of  $y=f(x)$  a distance  $c$  units upward.
• $y=f(x)-c$  shift the graph of  $y=f(x)$  a distance  $c$  units downward
• $y=f(x-c)$  shift the graph of  $y=f(x)$  a distance  $c$  to the right
• $y=f(x+c)$  shift the graph of  $y=f(x)$  a distance  $c$  to the left

Vertical and Horizotal Stretching and Reflecting: Suppose  $c> 1$  To obtain the graph of

• $y=cf(x)$  stretch the graph of  $y=f(x)$  vertically by a factor of  $c$
• $y=(\frac{1}{c})f(x)$  shrink the graph of   $y=f(x)$  vertically by a factor of  $c$
• $y=f(cx)$  shrink the graph of   $y=f(x)$  horizontally by a factor of  $c$
• $y=f(\frac{x}{c})$  stretch the graph of  $y=f(x)$  horizontally by a factor of  $c$
•  $y=-f(x)$  reflect the graph of  $y=f(x)$  about x-axis
• $y=f(-x)$  reflect the graph of  $y=f(x)$  about y-axis

Combination of function

Definition: Given two functions  $f$  and  $g$  the composite function  $\displaystyle f\circ g$ (also called the composition of  $f$  and  $g$  ) is defined by  $\displaystyle (f\circ g)(x)=f(g(x))$

Note: in general  $\displaystyle f\circ g\neq g\circ f$

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