Question

Determine whether the function shown in the graph is even or odd.

The graph starts at the bottom left, continues up through the x axis near negative one point seven five to a maximum around y equals two, goes back down through the x axis at the origin to a minimum around y equals negative two, and goes back up through the x axis near one point seven five.

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.

Answer 1

To determine if a function is even, odd, or neither based on its graph, we look for specific types of symmetry.

1. **Even functions** are symmetric with respect to the y-axis. This means that if we fold the graph along the y-axis, the two halves will match. Mathematically, a function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \).

2. **Odd functions** are symmetric with respect to the origin. This means that if we rotate the graph 180 degrees around the origin, it will look the same. Mathematically, a function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

From your description of the graph:
- The function crosses the x-axis at the origin (0,0).
- It reaches a maximum above the x-axis and then a minimum below it.
- Importantly, the graph appears to mirror itself across the origin based on the points you described.

Given these observations, the symmetry with respect to the origin suggests that the function is odd.

Thus, the correct conclusion is:
**The function is odd because it is symmetric with respect to the origin.**