Question

use your graphing calculator to sketch the graph of the function, and then determine the horizontal interval over which the function is decreasing.

$y = -x^2 + 96$

sketch the graph of the function in the viewing window $-20,20 \times -100,100$. choose the correct graph below.

\\( \bigcirc \\) a. \\( \bigcirc \\) b. \\( \bigcirc \\) c. \\( \bigcirc \\) d.

graphs of a, b, c, d are shown with magnifying and other icons

Explanation:

Step1: Analyze the function's form

The function \( y=-x^{2}+96 \) is a quadratic function in the form \( y = ax^{2}+bx + c \), where \( a=-1 \), \( b = 0 \), \( c = 96 \). For a quadratic function \( y=ax^{2}+bx + c \), if \( a<0 \), the parabola opens downward.

Step2: Find the vertex

The x - coordinate of the vertex of a quadratic function \( y = ax^{2}+bx + c \) is given by \( x=-\frac{b}{2a} \). Substituting \( a=-1 \) and \( b = 0 \), we get \( x = 0 \). Substituting \( x = 0 \) into the function \( y=-x^{2}+96 \), we get \( y=96 \). So the vertex of the parabola is at \( (0,96) \).

Step3: Compare with the given graphs

• Option A: The parabola opens upward, which does not match \( a=-1<0 \) (should open downward), so A is incorrect.
• Option B: The vertex of the parabola is at a negative y - value, but our vertex is at \( y = 96 \) (positive), so B is incorrect.
• Option C: The parabola opens downward and the vertex is at a positive y - value (consistent with \( y = 96 \)), so C is correct.
• Option D: The parabola opens upward, which does not match \( a=-1<0 \) (should open downward), so D is incorrect.

Answer:

C. The graph of \( y = -x^2 + 96 \) is a parabola opening downward (since the coefficient of \( x^2 \) is negative). The vertex form of a parabola is \( y = a(x - h)^2 + k \), here \( h = 0 \), \( k = 96 \), so the vertex is at \( (0, 96) \). Among the options, only option C shows a downward - opening parabola with vertex at a positive y - value (around 96 in the viewing window \([-20,20]\times[-100,100]\)).