Chapter 2, Problem 8CR

To determine

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

Answer to Problem 8CR

Vertex is $(-2,-25)$ .

The axis of symmetry is $x=-2$ .

The parabola opens up and has a minimum value of $y_{\min }=-25$ .

The function is decreasing on the interval $-\infty <x<-2$ and increasing on the interval $-2<x<\infty$ .

  

Explanation of Solution

Given information: $g(x)=(x-3)(x+7)$ .

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

From the given function $g(x)=(x-3)(x+7)$ ,

  $\begin{array}{l}g(x)=(x-3)(x+7)\\ g(x)=x^2+7x-3x-21\\ g(x)=x^2+4x-21\end{array}$

  $a=1,b=4,c=-21$

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

  $\begin{array}{l}x=-\frac{4}{2(1)}\\ x=-\frac{4}{2}\\ x=-2\\ \end{array}$

Substitute $x=-2$ in $y=x^2+4x-21$ and solve for $y$ ,

  $\begin{array}{l}$ y={(-2)}^2+4(-2)-21$y=4-8-21\\ y=-25\end{array}$

The vertex of the given function is $(h,k)$ is $(-2,-25)$ .

The vertical line passes through the $x-$ coordinate of the vertex is the axis of symmetry, $x=-2$ .

The given function opens up and has a minimum value of $y_{\min }=-25$ .

The graph of the given function is decreasing on the interval $-\infty <x<-2$ and increasing on the interval $-2<x<\infty$ .