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 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-values and y-values switch. The x -values and y -values switch. The x-intercepts become negated. The x -intercepts become negated. This function does not have any x-intercepts, so no changes are necessary.

Answer 1

To determine 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 \( y = 0 \):

\[
0 = x^2 - 25
\]

Solving for \( x \):

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

Thus, the x-intercepts of the function are at the points \( (5, 0) \) and \( (-5, 0) \).

Now, when the function \( y = x^2 - 25 \) is reflected over the x-axis, the new function becomes:

\[
y = -(x^2 - 25) = -x^2 + 25
\]

To find the new x-intercepts, we set \( y = 0 \) again:

\[
0 = -x^2 + 25
\]

Solving for \( x \):

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

The new x-intercepts remain \( (5, 0) \) and \( (-5, 0) \).

Since the x-intercepts are the points where the graph crosses the x-axis, which is determined solely by the x-values (where \( y = 0 \)), the x-intercepts do not change when the graph is reflected over the x-axis.

**Conclusion:** The x-intercepts remain the same because the y-values (which become negated) do not affect the x-values where the graph crosses the x-axis.

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