Question

use your graphing calculator to sketch the graph of the function, and then determine the x-interval over which the function is decreasing.

$g(x) = -x^2 + 13x - 40$

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.

graphs a, b, c, d are shown with zoom and other icons

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

(\square)

(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 \( g(x) = -x^2 + 13x - 40 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a=-1 \), \( b = 13 \), and \( c=-40 \). Since \( a=-1<0 \), the parabola opens downward.

Step2: Find the vertex's x - coordinate

The x - coordinate of the vertex of a quadratic function \( y = ax^2+bx + c \) is given by the formula \( x=-\frac{b}{2a} \).
Substitute \( a=-1 \) and \( b = 13 \) into the formula:
\( x=-\frac{13}{2\times(-1)}=\frac{13}{2}=6.5 \)

Step3: Determine the decreasing interval

For a parabola that opens downward (because \( a < 0 \)), the function is decreasing for all \( x \) - values greater than or equal to the x - coordinate of the vertex.
So the function \( g(x) \) is decreasing when \( x\geq6.5 \)

Answer:

$x \geq 6.5$