Question

use your graphing calculator to sketch the graph of the function, and then determine the x-interval over which the function is decreasing. g(x) = -x² + 15x - 54 sketch the graph of the function in the viewing window -10,10×-10,10. choose the correct graph below. ○ a. ○ b. ○ c. ○ d.

Explanation:

Step1: Analyze the function type

The function \( g(x) = -x^2 + 15x - 54 \) is a quadratic function. The general form of a quadratic function is \( y = ax^2 + bx + c \). Here, \( a = -1 \), \( b = 15 \), and \( c = -54 \). Since \( a = -1 < 0 \), the parabola opens downward.

Step2: Find the vertex of the parabola

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 \( b = 15 \) into the formula, we get \( x=-\frac{15}{2\times(-1)}=\frac{15}{2} = 7.5 \).

Step3: Analyze the graph in the window \([- 10,10]\times[-10,10]\)

Since the parabola opens downward and the vertex is at \( x = 7.5 \) (which is in the interval \([-10,10]\)), the graph should have a maximum point at \( x = 7.5 \). Among the given options, option B shows a parabola opening downward with the vertex in the right - hand side of the viewing window (around \( x = 7.5 \)), while option A has the vertex on the left, option C and D are upward - opening parabolas (since their graphs have a minimum point, which is characteristic of \( a>0 \)). So the correct graph is option B.

Answer:

B. (Assuming the graph in option B shows a parabola opening downward with vertex in the right half of the window, consistent with the function \( g(x) = -x^2 + 15x - 54 \))