Reflections
Learning Objectives
- Graph functions using reflections about the -axis and the -axis.
- Determine whether a function is even, odd, or neither from its graph.

A General Note: Reflections
Given a function , a new function is a vertical reflection of the function , sometimes called a reflection about (or over, or through) the x-axis. Given a function , a new function is a horizontal reflection of the function , sometimes called a reflection about the y-axis.How To: Given a function, reflect the graph both vertically and horizontally.
- Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis.
- Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis.
Example: Reflecting a Graph Horizontally and Vertically
Reflect the graph of (a) vertically and (b) horizontally.Answer: a. Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis as shown below.

Notice that this is an outside change, or vertical shift, that affects the output values, so the negative sign belongs outside of the function. b. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.

Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function. Note that these transformations can affect the domain and range of the functions. While the original square root function has domain and range , the vertical reflection gives the function the range and the horizontal reflection gives the function the domain .
Try It
Use Desmos to determine whether the following functions are even, odd, or neither:
Answer:
- even
- odd
- neither