What happens to the x-intercepts of the function y=x2−25

Question

when the graph is reflected over the x-axis? Explain.(1 point)
Responses

The x-intercepts become negated.
The x -intercepts become negated.

This function does not have any x-intercepts, so no changes are necessary.
This function does not have any x -intercepts, so no changes are necessary.

The x-values and y-values switch.
The x -values and y -values switch.

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

Answer 1

To analyze the effects of reflecting the function \( y = x^2 - 25 \) over the x-axis, we first need to identify the x-intercepts of the original function.

To find the x-intercepts, we set \( y = 0 \): \[ 0 = x^2 - 25 \] Solving this gives: \[ x^2 = 25 \implies x = 5 \quad \text{or} \quad x = -5 \] Thus, the x-intercepts of the function are \( (5, 0) \) and \( (-5, 0) \).

When the graph is reflected over the x-axis, the y-values of all points on the graph are negated. The reflection transformation can be represented as \( y = -f(x) \), so the new function after reflection will be: \[ y = -(x^2 - 25) = -x^2 + 25 \]

However, the x-intercepts are defined as the points where \( y = 0 \). The x-intercepts do not change when reflecting the graph over the x-axis, since their y-coordinates remain zero — they are on the x-axis. Therefore, the x-intercepts are still: \[ (5, 0) \quad \text{and} \quad (-5, 0) \]

Thus, the correct response is: The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.