Key Concepts & Glossary
Key Equations
Vertical shift | (up for ) |
Horizontal shift | (right for ) |
Vertical reflection | |
Horizontal reflection | |
Vertical stretch | ( ) |
Vertical compression | |
Horizontal stretch | |
Horizontal compression | ( ) |
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 axis. A graph can be reflected vertically by multiplying the output by –1.
- A horizontal reflection reflects a graph about the 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 axis, whereas odd functions are symmetric about the origin.
- Even functions satisfy the condition .
- Odd functions satisfy the condition .
- 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.
Glossary
- even function
- a function whose graph is unchanged by horizontal reflection, , and is symmetric about the axis
- horizontal compression
- a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant
- horizontal reflection
- a transformation that reflects a function’s graph across the y-axis by multiplying the input by
- horizontal shift
- a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
- horizontal stretch
- a transformation that stretches a function’s graph horizontally by multiplying the input by a constant
- odd function
- a function whose graph is unchanged by combined horizontal and vertical reflection, , and is symmetric about the origin
- vertical compression
- a function transformation that compresses the function’s graph vertically by multiplying the output by a constant
- vertical reflection
- a transformation that reflects a function’s graph across the x-axis by multiplying the output by
- vertical shift
- a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
- vertical stretch
- a transformation that stretches a function’s graph vertically by multiplying the output by a constant