Determine whether the function shown in the graph is even or odd.
The graph starts at the top left, continues down through the x axis at negative two to a minimum around y equals negative five, goes up to a maximum on the x axis at y equals negative one, goes back down to a minimum around y equals negative five, and goes back up through the x axis at two.

(03.03)
Determine whether the function shown in the graph is even or odd.
The graph starts at the top left, continues down through the x axis at negative two to a minimum around y equals negative five, goes up to a maximum on the x axis at y equals negative one, goes back down to a minimum around y equals negative five, and goes back up through the x axis at two.

The function is even because it is symmetric with respect to the y-axis.

The function is odd because it is symmetric with respect to the y-axis.

The function is even because it is symmetric with respect to the origin.

The function is odd because it is symmetric with respect to the origin.

To determine if a function is even or odd by examining its graph, we look at its symmetry:

1. **Even function**: A function is even if it is symmetric with respect to the y-axis. This means that if we reflect the graph across the y-axis, it would appear unchanged. Mathematically, an even function satisfies \( f(x) = f(-x) \) for all \( x \) in the domain.

2. **Odd function**: A function is odd if it is symmetric with respect to the origin. This means that if we rotate the graph 180 degrees around the origin, it would appear unchanged. Mathematically, an odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain.

From the description, the graph crosses the x-axis symmetrically at \( x = -2 \) and \( x = 2 \), with a similar pattern of minima and maxima on either side of the y-axis. This pattern indicates symmetry with respect to the origin (since it mirrors across both the x and y axes).

Thus, **the function is odd because it is symmetric with respect to the origin**.

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Would you like further clarification on symmetry in functions? Here are some related questions you may find helpful:

1. What is the general test to determine if a function is even or odd based on its equation?
2. Can a function be both even and odd?
3. How can we identify even and odd functions if the graph is less symmetrical?
4. How do transformations affect the evenness or oddness of a function?
5. Are all polynomial functions even or odd?

**Tip:** When analyzing functions visually, symmetry across the y-axis suggests an even function, while symmetry about the origin suggests an odd function.

Determine whether the function shown in the graph is even or odd. The graph starts at the top left, continues down through the x axis at negative two to a minimum around y equals negative five, goes up to a maximum on the x axis at y equals negative one, goes back down to a minimum around y equals negative five, and goes back up through the x axis at two.

Learn how to determine if a function is even or odd by analyzing its graph. This example includes symmetry concepts and key formulas for even and odd functions, ideal for high school students in Grades 9-12.

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