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 + 2x + 8 \\)\\( \dots \\)sketch the graph of the function in the viewing window \\( -10,10 \times -10,10 \\). choose the correct graph below.\\( \circ \\) a.\\( \circ \\) b.\\( \circ \\) c.\\( \circ \\) d.\\( \dots \\)find the x-interval, expressed as an inequality, over which the function is decreasing.\\( \square \\) (type an inequality or a compound inequality.)

Explanation:

Step1: Identify the function type

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

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 \(x=-\frac{b}{2a}\). Substituting \(a=-1\) and \(b = 2\) into the formula, we get \(x=-\frac{2}{2\times(-1)}=\frac{-2}{-2}=1\).

Step3: Determine the decreasing interval

For a parabola that opens downwards (because \(a < 0\)), the function is decreasing to the right of the vertex. So the function \(g(x)\) is decreasing for \(x\geq1\).

Answer:

\(x \geq 1\)