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) = -x² + 15x - 54

sketch the graph of the function in the viewing window -10,10×-10,10. choose the correct graph below.

○ a. ○ b. ○ c. ○ d.

find the x-interval, expressed as an inequality, over which the function is decreasing.
(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 \( g(x) = -x^2 + 15x - 54 \) is a quadratic function. The general form of a quadratic function is \( f(x)=ax^2+bx+c \), here \( a = - 1 \), \( b = 15 \), \( c=-54 \). Since \( a=-1<0 \), the parabola opens downwards.

Step2: Find the vertex of the parabola

The x - coordinate of the vertex of a quadratic function \( f(x)=ax^2 + bx + c \) is given by the formula \( x=-\frac{b}{2a} \).
Substitute \( a=-1 \) and \( b = 15 \) into the formula:
\( x=-\frac{15}{2\times(-1)}=\frac{15}{2} = 7.5 \)

Step3: Determine the interval where the function is decreasing

For a parabola that opens downwards ( \( a<0 \) ), the function is decreasing to the right of the vertex. So the function \( g(x) \) is decreasing for \( x\geq7.5 \) (or \( x > 7.5 \), but since we are talking about the interval where it is decreasing, including the vertex point as the boundary, we can write \( x\geq7.5 \)).

Answer:

\( x \geq 7.5 \)