1. Chapter 2
More on Functions

2. Section 2.1 Increasing, Decreasing, and Piecewise
Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry
2.5 Transformations
2.6 Variation and Applications

3. 2.4 Symmetry
Determine whether a graph is symmetric with respect to the x-axis, the y-axis, and the origin.
Determine whether a function is even, odd, or neither even nor odd.

4. Symmetry Algebraic Tests of Symmetry
x-axis: If replacing y with y produces an equivalent equation, then the graph is symmetric with respect to the x-axis.
y-axis: If replacing x with x produces an equivalent equation, then the graph is symmetric with respect to the y-axis.
Origin: If replacing x with x and y with y produces an equivalent equation, then the graph is symmetric with respect to the origin.

5. Example
Test y = x2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin.
x-axis: We replace y with y:
The resulting equation is not equivalent to the original so the graph is not symmetric with respect to the x-axis.

6. Example continued
Test y = x for symmetry with respect to the x-axis, the y-axis, and the origin.
y-axis: We replace x with x:
The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis.

7. Example continued Origin: We replace x with x and y with y:
The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.

8. Even and Odd Functions
If the graph of a function f is symmetric with respect to the y-axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f(x).
If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f(x) = f(x).

9. Example Determine whether the function is even, odd, or neither. 1.
We see that h(x) = h(x).
Thus, h is even.
y = x4  4x2

10. Example Determine whether the function is even, odd, or neither. 2.
y = x4  4x2
We see that h(x)  h(x). Thus, h is not odd.