Question

use your graphing calculator to sketch the graph of the function, and then determine the x-interval over which the function is increasing.\
\\( g(x) = -x^2 + 17x - 72 \\)\
sketch the graph of the function in the viewing window \\( -10,10 \times -10,10 \\). choose the correct graph below.\
\\( \circ \\) a.\
\\( \circ \\) b.\
\\( \circ \\) c.\
\\( \circ \\) d.

Explanation:

Step1: Analyze the function

The function \( g(x) = -x^2 + 17x - 72 \) is a quadratic function. The coefficient of \( x^2 \) is -1, so the parabola opens downwards.

Step2: Find the vertex x - coordinate

The x - coordinate of the vertex of a quadratic function \( ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). For \( g(x)=-x^{2}+17x - 72 \), \( a=- 1\) and \( b = 17 \). So \( x=-\frac{17}{2\times(-1)}=\frac{17}{2}=8.5 \).

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

Since the parabola opens downwards and the vertex is at \( x = 8.5\) (which is within the interval \([-10,10]\)), the graph should have a maximum at \( x = 8.5\) and open downwards. Among the given options, the graph that opens downwards and has a vertex in the right - hand side of the window (around \( x = 8.5\)) is option C.

Answer:

C. (assuming the third graph is the correct one based on the parabola opening downwards and vertex position)