Chapter 2, Problem 7CR

To determine

To label: The vertex and axis of symmetry, find the minimum and maximum value of the function, the function is increasing and decreasing.

Answer to Problem 7CR

Vertex is $(4,35)$ .

The axis of symmetry is $x=4$ .

The parabola opens up and has a maximum value of $y_{\max }=35$ .

The function is increasing on the interval $-\infty <x<4$ and decreasing on the interval $4<x<\infty$ .

  

Explanation of Solution

Given information: $g(x)=-2x^2+16x+3$ .

General form of a quadratic function is $y=a{x}^2+bx+c$ .

From the given function $g(x)=-2x^2+16x+3$ ,

  $a=-2,b=16,c=3$

The $x-$ coordinate of the vertex is given by the formula $x=-\frac{b}{2a}$ ,

  $\begin{array}{l}x=-\frac{16}{2(-2)}\\ x=-\frac{16}{-4}\\ x=4\\ \end{array}$

Substitute $x=4$ in $y=-2x^2+16x+3$ and solve for $y$ ,

  $\begin{array}{l}$ y=-2{(4)}^2+16(4)+3$y=-2(16)+16(4)+3\\ y=-32+64+3\\ y=25\end{array}$

The vertex of the given function is $(h,k)$ is $(4,35)$ .

The vertical line passes through the $x-$ coordinate of the vertex is the axis of symmetry, $x=4$ .

The given function opens up and has a maximum value of $y_{\max }=35$ .

The graph of the given function is increasing on the interval $-\infty <x<4$ and decreasing on the interval $4<x<\infty$ .