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.
(graphs are shown with options a, b, c, d, each with zoom and other icons)
find the x-interval, expressed as an inequality, over which the function is increasing.
(type an inequality or a compound inequality. use integers or decimals for any numbers in the inequality.)

Explanation:

Step1: Identify the function type

The function \(g(x)= -x^{2}+9x - 8\) is a quadratic function. The general form of a quadratic function is \(y = ax^{2}+bx + c\). Here, \(a=- 1\), \(b = 9\), \(c=-8\). Since \(a=-1<0\), the parabola opens downwards.

Step2: Find the vertex's x - coordinate

For a quadratic function \(y=ax^{2}+bx + c\), the x - coordinate of the vertex is given by the formula \(x=-\frac{b}{2a}\). Substitute \(a=-1\) and \(b = 9\) into the formula:
\(x=-\frac{9}{2\times(-1)}=\frac{9}{2}=4.5\)

Step3: Determine the increasing interval

Since the parabola opens downwards (because \(a < 0\)), the function is increasing to the left of the vertex. So the function \(g(x)\) is increasing when \(x<4.5\).

Answer:

\(x < 4.5\)