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

Question

when the graph is reflected over the x-axis? Explain.
Responses

The x-intercepts become negated.

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-values and y-values switch.

Answer 1

To analyze the function \( y = x^2 - 25 \) and its x-intercepts, we first find the x-intercepts by setting \( y = 0 \):

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

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

When we reflect this function over the x-axis, the new function will be:

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

In the reflected function, we still find the x-intercepts by setting \( y = 0 \):

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

We see that the x-intercepts remain \( (5, 0) \) and \( (-5, 0) \).

Thus, when reflecting the function over the x-axis, the x-intercepts remain the same because the y-values get negated during the reflection, but the x-values do not change.

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.