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

Question

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

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 become negated.
The x -intercepts become negated.

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

Answer 1

To analyze what happens to the x-intercepts of the function $y=x^2-25$ when the graph is reflected over the x-axis, we first determine the x-intercepts of the original function.

The x-intercepts occur where $y=0$:

$$0=x^2-25$$ $$x^2=25$$ $$x=5\text{or}x=-5$$

Thus, the x-intercepts of the function $y=x^2-25$ are at $(5,0)$ and $(-5,0)$.

Next, reflecting the graph over the x-axis modifies the function by negating the y-values. The new function after this reflection would be:

$$y=-(x^2-25)=-x^2+25$$

However, the x-intercepts depend only on where $y=0$. Let's set the new function to zero to find the x-intercepts of the reflected function:

$$0=-x^2+25$$ $$x^2=25$$ $$x=5\text{or}x=-5$$

This shows that the x-intercepts remain unchanged at $(5,0)$ and $(-5,0)$ after the reflection.

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