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-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. The x -intercepts remain the same because the y -values get negated during a reflection over the x -axis. The x-values and y-values switch.
Answers
GPT-4o mini
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 need to determine the x-intercepts of the original function.
The x-intercepts occur where \( y = 0 \):
\[
x^2 - 25 = 0
\]
Solving this equation:
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts of the function are at the points \( (5, 0) \) and \( (-5, 0) \).
Now, when we reflect the graph over the x-axis, the general transformation affects the y-coordinates of the points on the graph. Specifically, if the original point is \( (x, y) \), after reflection it becomes \( (x, -y) \).
For the x-intercepts, where \( y = 0 \), reflecting these points will produce new points:
1. For \( (5, 0) \): The reflected point is \( (5, -0) \) which is still \( (5, 0) \).
2. For \( (-5, 0) \): The reflected point is \( (-5, -0) \) which is still \( (-5, 0) \).
Since the y-coordinate remains zero for the x-intercepts after reflection, the x-intercepts do not change in position.
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 where \( y = 0 \):
\[
x^2 - 25 = 0
\]
Solving this equation:
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts of the function are at the points \( (5, 0) \) and \( (-5, 0) \).
Now, when we reflect the graph over the x-axis, the general transformation affects the y-coordinates of the points on the graph. Specifically, if the original point is \( (x, y) \), after reflection it becomes \( (x, -y) \).
For the x-intercepts, where \( y = 0 \), reflecting these points will produce new points:
1. For \( (5, 0) \): The reflected point is \( (5, -0) \) which is still \( (5, 0) \).
2. For \( (-5, 0) \): The reflected point is \( (-5, -0) \) which is still \( (-5, 0) \).
Since the y-coordinate remains zero for the x-intercepts after reflection, the x-intercepts do not change in position.
Thus, the correct response is:
**The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**