Question
What happens to the x-intercepts of the function y=x2−9 when the graph is reflected over the x-axis?(1 point) Responses There are no x-intercepts. There are no x-intercepts. The x-intercepts stay the same. The x-intercepts stay the same. The x-intercepts get shifted down 9 units. The x-intercepts get shifted down 9 units. The x-intercepts get shifted to the right 9 units.
Answers
GPT-4o mini
The function \( y = x^2 - 9 \) can be factored to find its x-intercepts (where \( y = 0 \)):
\[
x^2 - 9 = 0 \implies (x - 3)(x + 3) = 0 \implies x = 3 \text{ and } x = -3.
\]
Thus, the x-intercepts of the function are \( (3, 0) \) and \( (-3, 0) \).
When the graph is reflected over the x-axis, the function becomes:
\[
y = -(x^2 - 9) = -x^2 + 9.
\]
This new function can be analyzed to find the new x-intercepts by setting it equal to zero:
\[
-x^2 + 9 = 0 \implies x^2 = 9 \implies x = 3 \text{ and } x = -3.
\]
Therefore, reflecting the graph over the x-axis does not change the x-intercepts; they remain at \( (3, 0) \) and \( (-3, 0) \).
The correct answer is **The x-intercepts stay the same.**
\[
x^2 - 9 = 0 \implies (x - 3)(x + 3) = 0 \implies x = 3 \text{ and } x = -3.
\]
Thus, the x-intercepts of the function are \( (3, 0) \) and \( (-3, 0) \).
When the graph is reflected over the x-axis, the function becomes:
\[
y = -(x^2 - 9) = -x^2 + 9.
\]
This new function can be analyzed to find the new x-intercepts by setting it equal to zero:
\[
-x^2 + 9 = 0 \implies x^2 = 9 \implies x = 3 \text{ and } x = -3.
\]
Therefore, reflecting the graph over the x-axis does not change the x-intercepts; they remain at \( (3, 0) \) and \( (-3, 0) \).
The correct answer is **The x-intercepts stay the same.**