Question

What happens to the x-intercepts of the function y=x2−9
when the graph is reflected over the x-axis?(1 point)
Responses

There are no x-intercepts.
There are no x-intercepts.

The x-intercepts get shifted to the right 9 units.
The x-intercepts get shifted to the right 9 units.

The x-intercepts stay the same.
The x-intercepts stay the same.

The x-intercepts get shifted down 9 units.

Answer 1

To determine the effect on the x-intercepts of the function \( y = x^2 - 9 \) when the graph is reflected over the x-axis, we first need to find the x-intercepts of the original function.

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

\[
x^2 - 9 = 0
\]

This can be factored:

\[
(x - 3)(x + 3) = 0
\]

So the x-intercepts are \( x = 3 \) and \( x = -3 \).

When the graph is reflected over the x-axis, the new function becomes:

\[
y = -(x^2 - 9) = -x^2 + 9
\]

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

\[
-x^2 + 9 = 0
\]

Rearranging gives:

\[
x^2 = 9
\]

Taking the square root of both sides, we find:

\[
x = 3 \quad \text{and} \quad x = -3
\]

Thus, when the graph is reflected over the x-axis, the x-intercepts remain the same: \( x = 3 \) and \( x = -3 \).

Therefore, the correct response is:

**The x-intercepts stay the same.**