Key Concepts

• A function can be shifted vertically by adding a constant to the output.
• A function can be shifted horizontally by adding a constant to the input.
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
• Vertical and horizontal shifts are often combined.
• A vertical reflection reflects a graph about the $x\text{-}$ axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the $y\text{-}$ axis. A graph can be reflected horizontally by multiplying the input by –1.
• A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
• A function presented as an equation can be reflected by applying transformations one at a time.
• Even functions are symmetric about the $y\text{-}$ axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition $f\left(x\right)=f\left(-x\right)$.
• Odd functions satisfy the condition $f\left(x\right)=-f\left(-x\right)$.
• A function can be odd, even, or neither.
• A function can be compressed or stretched vertically by multiplying the output by a constant.
• A function can be compressed or stretched horizontally by multiplying the input by a constant.
• The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.