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) = -x² + 9x - 8 sketch the graph of the function in the viewing window -10,10×-15,15. choose the correct graph below. a. b. c. d.

Explanation:

Step1: Analyze the function type

The function \( g(x) = -x^2 + 9x - 8 \) is a quadratic function (\( ax^2 + bx + c \)) with \( a = -1 \), \( b = 9 \), \( c = -8 \). Since \( a < 0 \), the parabola opens downward.

Step2: Identify the correct graph

Look for the graph with a downward - opening parabola (since \( a=-1<0 \)). Options A and B have upward - opening parabolas (since they "open up"), while C and D have downward - opening parabolas. Now, we can further check the vertex. The x - coordinate of the vertex is \( x =-\frac{b}{2a}=-\frac{9}{2\times(-1)} = 4.5 \). In the viewing window \([-10,10]\times[-15,15]\), the vertex at \( x = 4.5 \) should be visible. The correct graph (with downward opening and vertex around \( x = 4.5 \)) is the appropriate one (e.g., if C is the one with downward opening and vertex at \( x = 4.5 \) in the window, then C is correct).

Step3: Determine the increasing interval

For a downward - opening parabola \( y = ax^2+bx + c\) (\( a<0 \)), the function is increasing on the interval \( (-\infty,-\frac{b}{2a}) \). Substituting \( a=-1 \) and \( b = 9 \), we get \( x<\frac{9}{2}=4.5 \). In the viewing window \([-10,10]\), the function is increasing on \([-10,4.5]\) (or \((-\infty,4.5)\) in general). But first, the graph choice: the correct graph is the one with downward opening, so we select the graph (e.g., option C) that matches the downward - opening parabola with vertex at \( x = 4.5 \).

Answer:

C. (Assuming the visual check: The function \( g(x) = -x^2 + 9x - 8 \) is a quadratic function with a negative leading coefficient, so it opens downward. The vertex of a parabola \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Here, \( a = -1 \), \( b = 9 \), so \( x = -\frac{9}{2(-1)} = \frac{9}{2} = 4.5 \). The graph should open downward, so we look for the graph with a downward opening parabola. Among the options, C (or the relevant one with downward opening) is correct. Then for the increasing interval, since it opens downward, it increases from \( (-\infty, 4.5) \) or in the window, from \( [-10, 4.5] \) approximately, but first the graph choice: the correct graph is the one with downward opening, so likely option C (depending on the visual, but the key is the parabola opens down as coefficient of \( x^2 \) is negative).)