Question

which statement describes the function g(x) = -9 + x^2?

a. the vertex of the function is (0, -9) and the zeros are (0, -3) and (0, 3)
b. the vertex of the function is (0, -9) and the zeros are (-3, 0) and (3, 0)
c. the vertex of the function is (0, 9) and the zeros are (0, -3) and (0, 3)
d. the vertex of the function is (0,9) and the zeros are (-3,0) and (3,0)

Answer 1

To analyze the function \( g(x) = -9 + x^2 \):

1. **Vertex:** The function is in the form of \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. This can be rearranged as:
\[
g(x) = x^2 - 9
\]
Here, \( h = 0 \) and \( k = -9 \), so the vertex is \((0, -9)\).

2. **Zeros (x-intercepts):** To find the zeros of the function, set \( g(x) = 0 \):
\[
-9 + x^2 = 0 \implies x^2 = 9 \implies x = \pm 3
\]
Therefore, the zeros are \((-3, 0)\) and \((3, 0)\).

From the analysis:
- The vertex is \((0, -9)\).
- The zeros are \((-3, 0)\) and \((3, 0)\).

Thus, the correct statement is:
**b. the vertex of the function is (0, -9) and the zeros are (-3, 0) and (3, 0).**