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

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How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 1


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%28x%29,

y = abs%28x%29%2B1,

y = -abs%28x%29%2B2.

This lesson has continuations in the lessons
    How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 2  and
    How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3,
where more complicated cases are considered:

y = abs%282x%2B3%29,

y = -abs%283x-6%29,

y = abs%282x%2B3%29 + x,

y = abs%282x%2B3%29 + abs%28x%29,

y = abs%28x%2B1%29 - abs%28x-1%29.


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

Let us start with the simplest case  y = abs%28x%29.


Problem 1

Plot the function  y  =  abs%28x%29.

Solution

You know that

    abs%28x%29 =    x,  if  x >= 0,  and                                                   
    abs%28x%29 =  -x,  if  x < 0.

Hence, the function  y  =  abs%28x%29  is

    y =    x,   if  x >= 0,  and
    y =  -x,  if  x < 0.

Thus, the entire set of real numbers is breaking up
into two domains,  x < 0  and  x >= 0,
and the function y = abs%28x%29  becomes

    y = -x  over the first domain  x < 0,  and
    y =   x   over the second domain  x >= 0.



Figure 1. Plot of the function  y = abs%28x%29

In  Figure 1  the green straight line represents the plot of the function  y = -x  over the domain  x < 0  and  the black straight line represents the plot of the function
y = x  over the domain  x >= 0.  Both straight lines together represent the plot of the function  y = abs%28x%29  over the entire set of real numbers.


Problem 2

Plot the function y = abs%28x%29%2B1.

Solution

Again, let us start from the definition

    abs%28x%29 =    x,   if  x >= 0,  and                                                   
    abs%28x%29 =  -x,  if  x < 0.

Hence, the function  y  =  abs%28x%29%2B1  is

    y =    x%2B1,  if  x >= 0,  and
    y =  -x%2B1,  if  x < 0.

Thus, the entire set of real numbers is breaking up
into two domains,  x < 0  and  x >= 0,
and the function y = abs%28x%29%2B1  becomes

    y = -x%2B1  over the first domain  x < 0,  and
    y =   x%2B1   over the second domain  x >= 0.



Figure 2. Plot of the function  y = abs%28x%29%2B1

In  Figure 2  the green straight line represents the plot of the function  y = -x%2B1  over the domain  x < 0  and  the black straight line represents the plot of the function
y = x%2B1  over the domain  x >= 0.  Both straight lines together represent the plot of the function  y = abs%28x%29%2B1  over the entire set of real numbers.


Problem 3

Plot the function y = -abs%28x%29%2B2.

Solution

By the definition

    abs%28x%29 =    x,   if  x >= 0,  and                                                   
    abs%28x%29 =  -x,  if  x < 0.

Hence, the function  y  =  -abs%28x%29%2B2  is

    y =  -x%2B2,   if  x >= 0,  and
    y =    x%2B2,    if  x < 0.

Thus, the entire set of real numbers is breaking up
into two domains,  x < 0  and  x >= 0,
and the function y = -abs%28x%29%2B2  becomes

    y =   x%2B2    over the first domain  x < 0,  and
    y = -x%2B2   over the second domain  x >= 0.



Figure 3. Plot of the function  y = -abs%28x%29%2B2

In  Figure 3  the green straight line represents the plot of the function  y = x%2B2  over the domain  x < 0  and  the black straight line represents the plot of the function
y = -x%2B2  over the domain  x >= 0.  Both straight lines together represent the plot of the function  y = -abs%28x%29%2B2  over the entire set of real numbers.


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




Figure 1. Plot of the function  y = abs%28x%29        



Figure 2. Plot of the function  y = abs%28x%29%2B1    



Figure 3. Plot of the function  y = -abs%28x%29%2B2


Note that the strategy is the same in all three examples.  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,  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
    How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 2
    How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3
    OVERVIEW of lessons on plotting functions with Linear Terms under the Absolute Value sign

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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