Chapter 4.7, Problem 51PPS

To determine

To Graph :

Graph the function.

Explanation of Solution

Given Information:

Given equation is :

  $y=f(x)=-4|x+2|-3$

Concept Used :

Absolute Value Function:

An absolute value function is a function that contains an algebraic expression within absolute value symbols. The absolute value of a number is its distance from 0 on the number line. Parent Function : $f(x)=\left|x\right|=\{\begin{array}{l}x;x>0\\ -x;x\le 0\end{array}$

  $\left|x\right|\ge 0$

Domain : All real numbers

Range : All non-negative real numbers

Graph :

   $f(x)=\left|x\right|$

Calculation :

We have :

  $y=f(x)=-4|x+2|-3$

  $\begin{array}{l}Since\\ \left|x+2\right|\ge 0\\ \Rightarrow -4\left|x+2\right|\le 0\\ \Rightarrow -4\left|x+2\right|-3\le 0-\mathrm{3..........................}[subtracting"3"each\_side]\\ \Rightarrow -4\left|x+2\right|-3\le -3\end{array}$

Make a table for different value in the domain :

  $y=f(x)=-4|x+2|-3=\{\begin{array}{l}-4(x+2)-3;x+2>0\\ 4(x+2)-3;x+2\le 0\end{array}=\{\begin{array}{l}-4x-11;x>-2\\ 4x+5;x\le -2\end{array}$

The center point of the absolute value graph is located where x +2 = 0. Use this x-value as the middle value in the values of the table.

x	$x+2$	$y=f(x)=-4|x+2|-3$
-4	-2	-11
-3	-1	-7
-2	0	-3
-1	1	-7
0	2	-11

Plotting the values and joining, we get the required graph :