Question

What happens to the x-intercepts of the function y=x2−25 when the graph is reflected over the x-axis? Explain.(1 point) Responses
A. The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis. B. The x-intercepts become negated.
C. This function does not have any x-intercepts, so no changes are necessary.
D. The x-values and y-values switch.

Answer 1

To determine what happens to the x-intercepts of the function \( y = x^2 - 25 \) when reflected over the x-axis, we need to understand the concept of x-intercepts and the impact of the reflection.

The x-intercepts occur when \( y = 0 \). For the function \( y = x^2 - 25 \):

1. Set \( y = 0 \):
\[
0 = x^2 - 25
\]
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
So, the x-intercepts are at \( (5, 0) \) and \( (-5, 0) \).

When reflecting a function over the x-axis, the effects can be described as follows:

- The reflection over the x-axis changes the sign of the y-values. Thus, for any point \( (x, y) \), after reflection, it becomes \( (x, -y) \).
- However, the x-intercepts, which are points where \( y = 0 \), will remain at \( (5, 0) \) and \( (-5, 0) \) because the y-value is zero.

Since reflecting the function does not affect the x-values at which the function equals zero, the x-intercepts remain unchanged.

Thus, the correct response is:

**A. The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**