Lesson HOW TO plot functions containing Linear Terms under the Absolute Value sign. Lesson 1

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<H2>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 1</H2>

In this lesson I will show you how to plot functions containing Linear terms under the Absolute value sign. 
This is the starting lesson. We consider, one after the other, the plotting procedure for the following simple examples:

{{{y}}} = {{{abs(x)}}},

{{{y}}} = {{{abs(x)+1}}},

{{{y}}} = {{{-abs(x)+2}}}.

This lesson has continuations in the lessons

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-2.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 2</A>&nbsp; and

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3</A>, 
where more complicated cases are considered:

{{{y}}} = {{{abs(2x+3)}}},

{{{y}}} = {{{-abs(3x-6)}}},

{{{y}}} = {{{abs(2x+3)}}} + {{{x}}},

{{{y}}} = {{{abs(2x+3)}}} + {{{abs(x)}}},

{{{y}}} = {{{abs(x+1)}}} - {{{abs(x-1)}}}.

When you learn how to construct these plots, you will easily proceed in many other cases.

Let us start with the simplest case &nbsp;{{{y}}} = {{{abs(x)}}}.

<B>Problem 1</B>

Plot the function &nbsp;{{{y}}}&nbsp; = &nbsp;{{{abs(x)}}}.

<B>Solution</B> 
<TABLE> 
  <TR>
  <TD>
You know that

&nbsp;&nbsp;&nbsp;&nbsp;{{{abs(x)}}} = &nbsp;&nbsp;&nbsp;{{{x}}}, &nbsp;if &nbsp;{{{x}}} >= 0, &nbsp;and &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  
&nbsp;&nbsp;&nbsp;&nbsp;{{{abs(x)}}} = &nbsp;{{{-x}}}, &nbsp;if &nbsp;{{{x}}} < 0.

Hence, the function &nbsp;{{{y}}}&nbsp; = &nbsp;{{{abs(x)}}} &nbsp;is

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;&nbsp;{{{x}}}, &nbsp;&nbsp;if &nbsp;{{{x}}} >= 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;{{{-x}}}, &nbsp;if &nbsp;{{{x}}} < 0.

Thus, the entire set of real numbers is breaking up 
into two domains, &nbsp;{{{x}}} < 0 &nbsp;and &nbsp;{{{x}}} >= 0, 
and the function {{{y}}} = {{{abs(x)}}} &nbsp;becomes

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = {{{-x}}} &nbsp;over the first domain &nbsp;{{{x}}} < 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;{{{x}}} &nbsp;&nbsp;over the second domain &nbsp;{{{x}}} >= 0. 
 </TD>
  <TD> 
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, 2.4, 0, 0)),
            line(0, 0, 2.4, 2.4)
)}}}

<B>Figure 1</B>. Plot of the function &nbsp;{{{y}}} = {{{abs(x)}}} 
  </TD>
  </TR>
</TABLE>
In &nbsp;<B>Figure 1</B>&nbsp; the green straight line represents the plot of the function &nbsp;{{{y}}} = {{{-x}}} &nbsp;over the domain &nbsp;{{{x}}} < 0 &nbsp;and&nbsp; the black straight line represents the plot of the function 
{{{y}}} = {{{x}}} &nbsp;over the domain &nbsp;{{{x}}} >= 0. &nbsp;Both straight lines together represent the plot of the function &nbsp;{{{y}}} = {{{abs(x)}}} &nbsp;over the entire set of real numbers.

<B>Problem 2</B>

Plot the function {{{y}}} = {{{abs(x)+1}}}.

<B>Solution</B>

<TABLE> 
  <TR>
  <TD>
Again, let us start from the definition

&nbsp;&nbsp;&nbsp;&nbsp;{{{abs(x)}}} = &nbsp;&nbsp;&nbsp;{{{x}}}, &nbsp;&nbsp;if &nbsp;{{{x}}} >= 0, &nbsp;and &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  
&nbsp;&nbsp;&nbsp;&nbsp;{{{abs(x)}}} = &nbsp;{{{-x}}}, &nbsp;if &nbsp;{{{x}}} < 0.

Hence, the function &nbsp;{{{y}}}&nbsp; = &nbsp;{{{abs(x)+1}}} &nbsp;is

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;&nbsp;{{{x+1}}}, &nbsp;if &nbsp;{{{x}}} >= 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;{{{-x+1}}}, &nbsp;if &nbsp;{{{x}}} < 0.

Thus, the entire set of real numbers is breaking up 
into two domains, &nbsp;{{{x}}} < 0 &nbsp;and &nbsp;{{{x}}} >= 0, 
and the function {{{y}}} = {{{abs(x)+1}}} &nbsp;becomes

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = {{{-x+1}}} &nbsp;over the first domain &nbsp;{{{x}}} < 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;{{{x+1}}} &nbsp;&nbsp;over the second domain &nbsp;{{{x}}} >= 0. 
 </TD>
  <TD> 
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, 3.4, 0, 1)),
            line(0, 1, 2.4, 3.4)
)}}}

<B>Figure 2</B>. Plot of the function &nbsp;{{{y}}} = {{{abs(x)+1}}} 
  </TD>
  </TR>
</TABLE>
In &nbsp;<B>Figure 2</B>&nbsp; the green straight line represents the plot of the function &nbsp;{{{y}}} = {{{-x+1}}} &nbsp;over the domain &nbsp;{{{x}}} < 0 &nbsp;and&nbsp; the black straight line represents the plot of the function 
{{{y}}} = {{{x+1}}} &nbsp;over the domain &nbsp;{{{x}}} >= 0. &nbsp;Both straight lines together represent the plot of the function &nbsp;{{{y}}} = {{{abs(x)+1}}} &nbsp;over the entire set of real numbers.

<B>Problem 3</B>

Plot the function {{{y}}} = {{{-abs(x)+2}}}.

<B>Solution</B>

<TABLE> 
  <TR>
  <TD>
By the definition

Hence, the function &nbsp;{{{y}}}&nbsp; = &nbsp;-{{{abs(x)+2}}} &nbsp;is

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;{{{-x+2}}}, &nbsp;&nbsp;if &nbsp;{{{x}}} >= 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;&nbsp;{{{x+2}}}, &nbsp;&nbsp;&nbsp;if &nbsp;{{{x}}} < 0.

Thus, the entire set of real numbers is breaking up 
into two domains, &nbsp;{{{x}}} < 0 &nbsp;and &nbsp;{{{x}}} >= 0, 
and the function {{{y}}} = {{{-abs(x)+2}}} &nbsp;becomes

&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = &nbsp;&nbsp;{{{x+2}}} &nbsp;&nbsp;&nbsp;over the first domain &nbsp;{{{x}}} < 0, &nbsp;and 
&nbsp;&nbsp;&nbsp;&nbsp;{{{y}}} = {{{-x+2}}} &nbsp;&nbsp;over the second domain &nbsp;{{{x}}} >= 0. 
 </TD>
  <TD> 
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, -0.4, 0, 2)),
            line(0, 2, 2.4, -0.4)
)}}}

<B>Figure 3</B>. Plot of the function &nbsp;{{{y}}} = {{{-abs(x)+2}}} 
  </TD>
  </TR>
</TABLE>
In &nbsp;<B>Figure 3</B>&nbsp; the green straight line represents the plot of the function &nbsp;{{{y}}} = {{{x+2}}} &nbsp;over the domain &nbsp;{{{x}}} < 0 &nbsp;and&nbsp; the black straight line represents the plot of the function 
{{{y}}} = {{{-x+2}}} &nbsp;over the domain &nbsp;{{{x}}} >= 0. &nbsp;Both straight lines together represent the plot of the function &nbsp;{{{y}}} = {{{-abs(x)+2}}} &nbsp;over the entire set of real numbers.

Below the plots of the <B>Figures 1</B>, <B>2</B> and <B>3</B> are presented again for your  convenience.

<TABLE> 
  <TR>
  <TD> 
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, 2.4, 0, 0)),
            line(0, 0, 2.4, 2.4)
)}}}

<B>Figure 1</B>. Plot of the function &nbsp;{{{y}}} = {{{abs(x)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
  </TD>
   <TD>
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, 3.4, 0, 1)),
            line(0, 1, 2.4, 3.4)
)}}}

<B>Figure 2</B>. Plot of the function &nbsp;{{{y}}} = {{{abs(x)+1}}}&nbsp;&nbsp;&nbsp;&nbsp; 
  </TD>
   <TD>
{{{drawing( 250, 250, -2.5, 2.5, -1.5, 3.5,
            grid(1),
      green(line (-2.4, -0.4, 0, 2)),
            line(0, 2, 2.4, -0.4)
)}}}

<B>Figure 3</B>. Plot of the function &nbsp;{{{y}}} = {{{-abs(x)+2}}} 
  </TD>
  </TR>
</TABLE>

Note that the strategy is the same in all three examples. &nbsp;It is to break up the entire set of real numbers into sub-domains (ranges) 
where the absolute value of linear term is a linear function, &nbsp;and then to plot all these piece-wise linear functions.

My other lessons in this site on plotting functions containing linear terms under the Absolute Value sign are

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3</A>

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/absolute-value/Review-of-lessons-on-plotting-functions-with-Linear-Terms-under-the-Abs-Value-sign.lesson>OVERVIEW of lessons on plotting functions with Linear Terms under the Absolute Value sign</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.