Question

a quadratic function is shown.
g(x) = -(x - 3)² + 4
which statements are true about the graph of the function g.
select three correct answers.
use desmos graphing calculator for assignment do not use or open any other internet browser tabs.
(desmos graphing calculator interface shown with a coordinate plane)
options:
f. the vertex of the graph is at (- 3, - 4).
b. the function has a zero at - 5.
c. the minimum value of the function is 4.
e. the vertex of the graph is at (3, 4).
d. the maximum value of the function is 4.
a. the function has zeros at 1 and 5.

Explanation:

1. Analyze the vertex form of the quadratic function \( g(x) = -(x - 3)^2 + 4 \). The vertex form of a quadratic is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( h = 3 \) and \( k = 4 \), so the vertex is \((3, 4)\) (supports option E).
2. The coefficient of \((x - 3)^2\) is \(-1\), which is negative. So the parabola opens downward, meaning it has a maximum value at the vertex. The maximum value is \( k = 4 \) (supports option D).
3. Check for zeros: Set \( g(x) = 0 \), so \( -(x - 3)^2 + 4 = 0 \). Then \((x - 3)^2 = 4\), \( x - 3 = \pm 2 \), so \( x = 3 + 2 = 5 \) or \( x = 3 - 2 = 1 \). Thus, the function has zeros at \( 1 \) and \( 5 \) (supports option A).
4. Option B: The function opens downward, so it has a maximum, not a zero at \(-5\) (incorrect).
5. Option C: The function has a maximum (since \( a < 0 \)), not a minimum (incorrect).
6. Option F: The vertex is \((3, 4)\), not \((-3, -4)\) (incorrect).

Answer:

A. The function has zeros at 1 and 5,
D. The maximum value of the function is 4,
E. The vertex of the graph is at (3, 4)