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.

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

(images of graphs labeled a, b, c, d are shown with zoom and other icons)

find the x-interval, expressed as an inequality, over which the function is increasing.

(text box for answer: 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 \(g(x)=-x^{2}+17x - 72\) is a quadratic function in the form \(y = ax^{2}+bx + c\), where \(a=- 1\), \(b = 17\), \(c=-72\). 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 \(b = 17\) into the formula:
\(x=-\frac{17}{2\times(-1)}=\frac{17}{2}=8.5\)

Step3: Determine the increasing interval

For a parabola that opens downwards (because \(a < 0\)), the function is increasing to the left of the vertex and decreasing to the right of the vertex. So the function \(g(x)\) is increasing when \(x<8.5\).

Answer:

\(x < 8.5\)