Question
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)
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.
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 remain the same because the y-values get negated during a reflection over the x-axis.
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.
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 remain the same because the y-values get negated during a reflection over the x-axis.
Answers
GPT-4o mini
To determine what happens to the x-intercepts of the function \( y = x^2 - 25 \) when the graph is reflected over the x-axis, let's first find the x-intercepts.
The x-intercepts occur when \( y = 0 \):
\[
x^2 - 25 = 0
\]
Solving this equation, we get:
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
So the x-intercepts are at \( x = 5 \) and \( x = -5 \).
Now, reflecting the graph over the x-axis changes the sign of all y-values, transforming the function to \( y = -(x^2 - 25) = -x^2 + 25 \). However, this reflection does not affect the x-intercepts themselves. The x-intercepts are the points where the function intersects the x-axis, and this remains unchanged because the condition for x-intercepts (setting \( y = 0 \)) does not change.
Thus, the correct response is:
**The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**
The x-intercepts occur when \( y = 0 \):
\[
x^2 - 25 = 0
\]
Solving this equation, we get:
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
So the x-intercepts are at \( x = 5 \) and \( x = -5 \).
Now, reflecting the graph over the x-axis changes the sign of all y-values, transforming the function to \( y = -(x^2 - 25) = -x^2 + 25 \). However, this reflection does not affect the x-intercepts themselves. The x-intercepts are the points where the function intersects the x-axis, and this remains unchanged because the condition for x-intercepts (setting \( y = 0 \)) does not change.
Thus, the correct response is:
**The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**