Graphing Reflections and Translations

Reflections

1. Across the Y-Axis
   For any $a>1,\:f\left(x\right)=a^x\:$and $g\left(x\right)=\left(\frac{1}{a}\right)^x$are reflections across the y-axis. This is because $g\left(x\right)=\left(\frac{1}{a}\right)^x=\left(a^{-1}\right)^x=a^{-x}=f\left(-x\right)$.
   
2. Across the x-axis
   The graphs of $f\left(x\right)=a^x\:$and $g\left(x\right)=-a^x\:$are reflections across the x-axis.Here is an example with graphs of $y=2^x\:$and $f\left(x\right)=-2^x.$
   
   

Translations

1. Left or Right
   The graph of $f\left(x\right)=a^x\:$will be translated c units resulting in the graph of $g\left(x\right)=a^{x+c},\:$if $f\left(x\right)=a^x\:$translates to the left, or $k\left(x\right)=a^{x-c},\:$if translating c units to the right.
   For example, here is a graph of $y=2^x\:$and $f\left(x\right)=2^{x+3}\:$showing how replacing the$x$ with $x+3$ moves the graph to the left 3 units.
   
2. Up or Down
   The graph of $f\left(x\right)=a^x\:$will move k units when $g\left(x\right)=a^x+k\:$is graphed. It will move up for $k>0,\:$and down for $k<0.$
   For example, here is a graph of $y=2^x\:$along with a graph of $f\left(x\right)=2^{x-2}-1.$ Notice how the original graph of $y=2^x\:$is moved 2 units to the right (because the $x$ was replaced with $x-2$) and 1 unit down (because of the addition of $-1$ at the end).