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

Question

𝑦
=
𝑥
2
−
25
when the graph is reflected over the x-axis? Explain.(1 point)
Responses

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

The x-intercepts become negated.
The x -intercepts become negated.

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.

Answer 1

The correct response is:

The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.

Here's the reasoning behind it:

1. The x-intercepts of a function are the points where the graph intersects the x-axis, which occurs when \(y = 0\).

2. For the function \(y = x^2 - 25\), the x-intercepts are found by setting the equation to 0: \[ x^2 - 25 = 0 \] This factors to: \[ (x - 5)(x + 5) = 0 \] So, the x-intercepts are \(x = 5\) and \(x = -5\).

3. When the graph is reflected over the x-axis, the equation of the function changes to \(y = - (x^2 - 25)\) or \(y = -x^2 + 25\).

4. In both cases, the x-intercepts remain the same (at \(x = 5\) and \(x = -5\)) because the x-intercepts are determined by the values of x that make y = 0, regardless of the sign of the values of y. Therefore, reflecting the graph over the x-axis only negates the y-values across the axis, not the x-values where the graph intersects the x-axis.