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 down 9 units.
The x-intercepts get shifted down 9 units.

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.

Answer 1

To find the x-intercepts of the function \( y = x^2 - 9 \), we set \( y = 0 \):

\[
0 = x^2 - 9 \\
x^2 = 9 \\
x = 3 \quad \text{or} \quad x = -3
\]

So, the x-intercepts are at \( (3, 0) \) and \( (-3, 0) \).

Now, when the graph is reflected over the x-axis, we change the function to \( y = - (x^2 - 9) = -x^2 + 9 \).

The new equation is \( y = -x^2 + 9 \). To find the new x-intercepts, we again set \( y = 0 \):

\[
0 = -x^2 + 9 \\
x^2 = 9 \\
x = 3 \quad \text{or} \quad x = -3
\]

Thus, reflecting the graph over the x-axis does not change the x-intercepts; they remain at \( (3, 0) \) and \( (-3, 0) \).

Therefore, the correct response is:

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