Question

A) What happens to the x-intercepts of the function y = x
2 − 25 when the graph is reflected over
the x-axis? Explain. (1 point)
This function does not have any x-intercepts, so no changes are necessary.
The x-intercepts become negated.
The x-values and y-values switch.
The x-intercepts remain the same because the y-values get negated during a reflection over the xaxis

Answer 1

In the case of the function \( y = x^2 - 25 \), the x-intercepts occur when \( y = 0 \). To find the x-intercepts, we set the equation to zero:

\[
x^2 - 25 = 0
\]

Solving this gives:

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

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

When the graph is reflected over the x-axis, the y-values are negated, so the new function becomes:

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

Now, we assess the new function to find its x-intercepts:

\[
-x^2 + 25 = 0
\]

This simplifies to:

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

The x-intercepts remain the same, at \( (5, 0) \) and \( (-5, 0) \).

Therefore, the correct explanation is:

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