Question
What happens to the x-intercepts of the function y=x2−25
when the graph is reflected over the x-axis? Explain.(1 point)
Responses
The x-intercepts become negated.
The x -intercepts become negated.
The x-values and y-values switch.
The x -values and y -values switch.
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.
when the graph is reflected over the x-axis? Explain.(1 point)
Responses
The x-intercepts become negated.
The x -intercepts become negated.
The x-values and y-values switch.
The x -values and y -values switch.
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.
Answers
GPT-4o mini
To determine the effect of reflecting the function \( y = x^2 - 25 \) over the x-axis, we need to analyze the concept of x-intercepts. The x-intercepts are found by setting \( y = 0 \):
\[
0 = x^2 - 25
\]
Solving for \( x \) gives:
\[
x^2 = 25 \\
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts of the function are at \( x = 5 \) and \( x = -5 \).
When we reflect the graph of a function over the x-axis, the new function becomes \( y = -f(x) \). For our function, this results in:
\[
y = -(x^2 - 25) = -x^2 + 25
\]
The x-intercepts of the new function are found by setting \( y = 0 \):
\[
0 = -x^2 + 25 \\
x^2 = 25 \\
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts remain the same at \( x = 5 \) and \( x = -5 \) after the reflection.
Therefore, the correct response is:
**The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**
\[
0 = x^2 - 25
\]
Solving for \( x \) gives:
\[
x^2 = 25 \\
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts of the function are at \( x = 5 \) and \( x = -5 \).
When we reflect the graph of a function over the x-axis, the new function becomes \( y = -f(x) \). For our function, this results in:
\[
y = -(x^2 - 25) = -x^2 + 25
\]
The x-intercepts of the new function are found by setting \( y = 0 \):
\[
0 = -x^2 + 25 \\
x^2 = 25 \\
x = 5 \quad \text{or} \quad x = -5
\]
Thus, the x-intercepts remain the same at \( x = 5 \) and \( x = -5 \) after the reflection.
Therefore, the correct response is:
**The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.**