What happens to the x-intercepts of the function y=x2−25 when the graph is reflected over the x-axis? Explain.(1 point)

Question

Responses

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

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

The x-intercepts become negated.

Answer 1

To understand what happens to the x-intercepts of the function \( y = x^2 - 25 \) when the graph is reflected over the x-axis, we first need to identify the x-intercepts of the original function.

The x-intercepts occur where \( y = 0 \):

\[ 0 = x^2 - 25 \] \[ x^2 = 25 \] \[ x = 5 \quad \text{or} \quad x = -5 \]

So, the x-intercepts of the function \( y = x^2 - 25 \) are at \( x = 5 \) and \( x = -5 \).

When the graph is reflected over the x-axis, the new function would be \( y = -(x^2 - 25) = -x^2 + 25 \).

In this new function, the x-intercepts are found by setting \( y = 0 \) again:

\[ 0 = -x^2 + 25 \] \[ x^2 = 25 \] \[ x = 5 \quad \text{or} \quad x = -5 \]

Thus, the x-intercepts remain at \( x = 5 \) and \( x = -5 \).

The correct response to the question is:

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