absolute value functions assignment

3.6 Absolute Value Functions

Learning objectives.

In this section, you will:

• Graph an absolute value function.
• Solve an absolute value equation.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions .

Understanding Absolute Value

Recall that in its basic form f ( x ) = | x | , f ( x ) = | x | , the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

Absolute Value Function

The absolute value function can be defined as a piecewise function

Using Absolute Value to Determine Resistance

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%, ± 5%, ±1%, ± 5%, or ± 10% . ± 10% .

Suppose we have a resistor rated at 680 ohms, ± 5 % . ± 5 % . Use the absolute value function to express the range of possible values of the actual resistance.

We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance R R in ohms,

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 2 .

Figure 3 shows the graph of y = 2 | x – 3 | + 4. y = 2 | x – 3 | + 4. The graph of y = | x | y = | x | has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at ( 3 , 4 ) ( 3 , 4 ) for this transformed function.

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4 .

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5 .

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 6 .

From this information we can write the equation

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative, there is also a reflection about the x-axis.

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x x and f ( x ) . f ( x ) .

Now substituting in the point (1, 2)

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7 ).

Solving an Absolute Value Equation

In Other Type of Equations , we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 8 = | 2 x − 6 | , 8 = | 2 x − 6 | , we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

Solutions to Absolute Value Equations

For real numbers A A and B B , an equation of the form | A | = B , | A | = B , with B ≥ 0 , B ≥ 0 , will have solutions when A = B A = B or A = − B . A = − B . If B < 0 , B < 0 , the equation | A | = B | A | = B has no solution.

Given the formula for an absolute value function, find the horizontal intercepts of its graph .

• Isolate the absolute value term.
• Use | A | = B | A | = B to write A = B A = B or −A = B , −A = B , assuming B > 0. B > 0.
• Solve for x . x .

Finding the Zeros of an Absolute Value Function

For the function f ( x ) = | 4 x + 1 | − 7 , f ( x ) = | 4 x + 1 | − 7 , find the values of x x such that f ( x ) = 0. f ( x ) = 0.

The function outputs 0 when x = 3 2 x = 3 2 or x = − 2. x = − 2. See Figure 8 .

For the function f ( x ) = | 2 x − 1 | − 3 , f ( x ) = | 2 x − 1 | − 3 , find the values of x x such that f ( x ) = 0. f ( x ) = 0.

Should we always expect two answers when solving | A | = B ? | A | = B ?

No. We may find one, two, or even no answers. For example, there is no solution to 2 + | 3 x − 5 | = 1. 2 + | 3 x − 5 | = 1.

Access these online resources for additional instruction and practice with absolute value.

• Graphing Absolute Value Functions
• Graphing Absolute Value Functions 2

3.6 Section Exercises

How do you solve an absolute value equation?

How can you tell whether an absolute value function has two x -intercepts without graphing the function?

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

How can you use the graph of an absolute value function to determine the x -values for which the function values are negative?

Describe all numbers x x that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Describe all numbers x x that are at a distance of 1 2 1 2 from the number −4. Express this set of numbers using absolute value notation.

Describe the situation in which the distance that point x x is from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Find all function values f ( x ) f ( x ) such that the distance from f ( x ) f ( x ) to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

For the following exercises, find the x - and y -intercepts of the graphs of each function.

f ( x ) = 4 | x − 3 | + 4 f ( x ) = 4 | x − 3 | + 4

f ( x ) = − 3 | x − 2 | − 1 f ( x ) = − 3 | x − 2 | − 1

f ( x ) = − 2 | x + 1 | + 6 f ( x ) = − 2 | x + 1 | + 6

f ( x ) = − 5 | x + 2 | + 15 f ( x ) = − 5 | x + 2 | + 15

f ( x ) = 2 | x − 1 | − 6 f ( x ) = 2 | x − 1 | − 6

f ( x ) = | − 2 x + 1 | − 13 f ( x ) = | − 2 x + 1 | − 13

f ( x ) = − | x − 9 | + 16 f ( x ) = − | x − 9 | + 16

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

y = | x − 1 | y = | x − 1 |

y = | x + 1 | y = | x + 1 |

y = | x | + 1 y = | x | + 1

For the following exercises, graph the given functions by hand.

y = | x | − 2 y = | x | − 2

y = − | x | y = − | x |

y = − | x | − 2 y = − | x | − 2

y = − | x − 3 | − 2 y = − | x − 3 | − 2

f ( x ) = − | x − 1 | − 2 f ( x ) = − | x − 1 | − 2

f ( x ) = − | x + 3 | + 4 f ( x ) = − | x + 3 | + 4

f ( x ) = 2 | x + 3 | + 1 f ( x ) = 2 | x + 3 | + 1

f ( x ) = 3 | x − 2 | + 3 f ( x ) = 3 | x − 2 | + 3

f ( x ) = | 2 x − 4 | − 3 f ( x ) = | 2 x − 4 | − 3

f ( x ) = | 3 x + 9 | + 2 f ( x ) = | 3 x + 9 | + 2

f ( x ) = − | x − 1 | − 3 f ( x ) = − | x − 1 | − 3

f ( x ) = − | x + 4 | − 3 f ( x ) = − | x + 4 | − 3

f ( x ) = 1 2 | x + 4 | − 3 f ( x ) = 1 2 | x + 4 | − 3

Use a graphing utility to graph f ( x ) = 10 | x − 2 | f ( x ) = 10 | x − 2 | on the viewing window [ 0 , 4 ] . [ 0 , 4 ] . Identify the corresponding range. Show the graph.

Use a graphing utility to graph f ( x ) = − 100 | x | + 100 f ( x ) = − 100 | x | + 100 on the viewing window [ − 5 , 5 ] . [ − 5 , 5 ] . Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3 f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3

f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9 f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9

For the following exercises, solve the inequality.

If possible, find all values of a a such that there are no x - x - intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

If possible, find all values of a a such that there are no y y -intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

Real-World Applications

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x x represents the distance from city B to city A, express this using absolute value notation.

The true proportion p p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x x for the score.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x x as the diameter of the bearing, write this statement using absolute value notation.

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x x inches, express the tolerance using absolute value notation.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/3-6-absolute-value-functions

© Jan 9, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Chapter 3 Functions

3.7 Absolute Value Functions

Learning objectives.

In this section, you will:

• Graph an absolute value function.
• Solve an absolute value equation.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light-years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions .

Understanding Absolute Value

Recall that in its basic form [latex]\,f\left(x\right)=|x|,\,[/latex] the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

Absolute Value Function

The absolute value function can be defined as a piecewise function:

Using Absolute Value to Determine Resistance

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often[latex]\,\text{±1%,}\,±\text{5%,}\,[/latex]or[latex]\,±\text{10%}\text{.}[/latex]

Suppose we have a resistor rated at 680 ohms,[latex]\,±\text{5%}\,[/latex].  Use the absolute value function to express the range of possible values of the actual resistance.

We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so with the resistance [latex]\,R\,[/latex] in ohms,

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Using the variable[latex]\,p\,[/latex] for passing, [latex]\,|p-80|\le 20[/latex]

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 2.

Figure 3 shows the graph of [latex]\,y=2|x–3|+4.\,[/latex] The graph of [latex]\,y=|x|\,[/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\,\left(3,4\right)\,[/latex] for this transformed function.

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4.

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5.

We also notice that the graph appears vertically stretched because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance, as shown in Figure 6.

From this information, we can write the equation

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]\,x\,[/latex] and [latex]\,f\left(x\right).[/latex]

Now substituting in the point (1, 2)

Write the equation for the absolute value function that is horizontally shifted left 2 units, vertically flipped, and vertically shifted up 3 units.

[latex]f\left(x\right)=-|x+2|+3[/latex]

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7).

Access these online resources for additional instruction and practice with graphing absolute value equations.

Solving an Absolute Value Equation

In Other Type of Equations , we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]\,8=|2x-6|,\,[/latex] we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

Solutions to Absolute Value Equations

For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B,[/latex] with [latex]B\ge 0,[/latex] will have solutions when [latex]A=B[/latex] or [latex]A=-B.[/latex] If B<0 , the equation [latex]|A|=B[/latex] has no solution.

Given the formula for an absolute value function, find the horizontal intercepts of its graph .

• Isolate the absolute value term.
• Use [latex]\,|A|=B\,[/latex] to write [latex]\,A=B\,[/latex] or [latex]\,\mathrm{-A}=B,\,[/latex] assuming [latex]\,B>0.[/latex]
• Solve for [latex]\,x.\,[/latex]

Finding the Zeros of an Absolute Value Function

For the function[latex]\,f\left(x\right)=|4x+1|-7,[/latex] find the values of[latex]\,x\,[/latex]such that[latex]\,f\left(x\right)=0.[/latex]

The function outputs 0 when[latex]\,x=\frac{3}{2}\,[/latex]or[latex]\,x=-2.[/latex] See Figure 8.

For the function [latex]\,f\left(x\right)=|2x-1|-3,[/latex] find the values of [latex]\,x\,[/latex] such that [latex]\,f\left(x\right)=0.[/latex]

[latex]x=-1\,[/latex] or [latex]\,\,x=2[/latex]

Should we always expect two answers when solving[latex]\,|A|=B?[/latex]

No. We may find one, two, or even no answers. For example, there is no solution to  [latex]\,2+|3x-5|=1.[/latex]

Key Concepts

• Applied problems, such as ranges of possible values, can also be solved using the absolute value function.
• The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.
• In an absolute value equation, an unknown variable is the input of an absolute value function.
• If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.

Section Exercises

• How do you solve an absolute value equation?

Isolate the absolute value term so that the equation is of the form[latex]\,|A|=B.\,[/latex]Form one equation by setting the expression inside the absolute value symbol,[latex]\,A,\,[/latex]equal to the expression on the other side of the equation,[latex]\,B.\,[/latex]Form a second equation by setting[latex]\,A\,[/latex]equal to the opposite of the expression on the other side of the equation,[latex]\,-B.\,[/latex]Solve each equation for the variable.

• How can you tell whether an absolute value function has two x -intercepts without graphing the function?
• When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

The graph of the absolute value function does not cross the[latex]\,x[/latex]-axis, so the graph is either completely above or completely below the[latex]\,x[/latex]-axis.

• How can you use the graph of an absolute value function to determine the x -values for which the function values are negative?
• Describe all numbers[latex]\,x\,[/latex]that are at a distance of[latex]\,\frac{1}{2}\,[/latex]from the number −4. Express this set of numbers using absolute value notation.

[latex]\,|x+4|=\frac{1}{2}\,[/latex]

• Describe the situation in which the distance that point[latex]\,x\,[/latex]is from 10 is at least 15 units. Express this set of numbers using absolute value notation.
• Find all function values[latex]\,f\left(x\right)\,[/latex]such that the distance from[latex]\,f\left(x\right)\,[/latex]to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

[latex]|f\left(x\right)-8|[/latex] <0.03

For the following exercises, find the x – and y -intercepts of the graphs of each function.

• [latex]f\left(x\right)=4|x-3|+4[/latex]
• [latex]f\left(x\right)=-3|x-2|-1[/latex]

[latex]\left(0,-7\right);\,[/latex]no[latex]\,x[/latex]-intercepts

• [latex]f\left(x\right)=-2|x+1|+6[/latex]
• [latex]f\left(x\right)=-5|x+2|+15[/latex]

[latex]\left(0,\,5\right),\left(1,0\right),\left(-5,0\right)[/latex]

• [latex]f\left(x\right)=2|x-1|-6[/latex]
• [latex]f\left(x\right)=|-2x+1|-13[/latex]

[latex]\left(0,-12\right),\left(-6,0\right),\left(7,0\right)[/latex]

• [latex]f\left(x\right)=-|x-9|+16[/latex]

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

• [latex]y=|x-1|[/latex]

• [latex]y=|x+1|[/latex]
• [latex]y=|x|+1[/latex]

For the following exercises, graph the given functions by hand.

• [latex]y=|x|-2[/latex]
• [latex]y=-|x|[/latex]

• [latex]y=-|x|-2[/latex]
• [latex]y=-|x-3|-2[/latex]

• [latex]f\left(x\right)=-|x-1|-2[/latex]
• [latex]f\left(x\right)=-|x+3|+4[/latex]

• [latex]f\left(x\right)=2|x+3|+1[/latex]
• [latex]f\left(x\right)=3|x-2|+3[/latex]

• [latex]f\left(x\right)=|2x-4|-3[/latex]
• [latex]f\left(x\right)=|3x+9|+2[/latex]

• [latex]f\left(x\right)=-|x-1|-3[/latex]
• [latex]f\left(x\right)=-|x+4|-3[/latex]

• [latex]f\left(x\right)=\frac{1}{2}|x+4|-3[/latex]
• Use a graphing utility to graph [latex]f\left(x\right)=10|x-2|[/latex] on the viewing window [latex]\left[0,4\right].[/latex] Identify the corresponding range. Show the graph.

range:[latex]\,\left[0,20\right][/latex]

• Use a graphing utility to graph[latex]\,f\left(x\right)=-100|x|+100\,[/latex]on the viewing window[latex]\,\left[-5,5\right].\,[/latex]Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

• [latex]f\left(x\right)=-0.1|0.1\left(0.2-x\right)|+0.3[/latex]

[latex]x\text{-}[/latex]intercepts:

• [latex]f\left(x\right)=4×{10}^{9}|x-\left(5×{10}^{9}\right)|+2×{10}^{9}[/latex]

For the following exercises, solve the inequality.

• If possible, find all values of[latex]\,a\,[/latex]such that there are no [latex]\,y[/latex]-intercepts for[latex]\,f\left(x\right)=2|x+1|+a.[/latex]

There is no solution for[latex]\,a\,[/latex]that will keep the function from having a[latex]\,y[/latex]-intercept. The absolute value function always crosses the [latex]\,y[/latex]-intercept when[latex]\,x=0.[/latex]

• If possible, find all values of [latex]a[/latex] such that there are no [latex]x\text{-}[/latex]intercepts for [latex]f\left(x\right)=2|x+1|+a.[/latex]

Real-World Applications

• The true proportion[latex]\,p\,[/latex]of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

[latex]|p-0.08|\le 0.015[/latex]

• Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable[latex]\,x\,[/latex]for the score.
• A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using[latex]\,x\,[/latex]as the diameter of the bearing, write this statement using absolute value notation.

[latex]|x-5.0|\le 0.01[/latex]

• The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is[latex]\,x\,[/latex]inches, express the tolerance using absolute value notation.

Media Attributions

• 3.7 Figure 1 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 2 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 3 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 4 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 5 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 6 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 7 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Figure 8 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
• 3.7 Solution-15 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-17 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-19 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-21 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-23 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-25 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-27 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-29 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-31 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 3.7 Solution-33 © OpenStax College Algebra and Trigonometry is licensed under a CC BY (Attribution) license

College Algebra Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

Share This Book

Child Login

• Kindergarten
• Number charts
• Skip Counting
• Place Value
• Number Lines
• Subtraction
• Multiplication
• Word Problems
• Comparing Numbers
• Ordering Numbers
• Odd and Even
• Prime and Composite
• Roman Numerals
• Ordinal Numbers
• In and Out Boxes
• Number System Conversions
• More Number Sense Worksheets
• Size Comparison
• Measuring Length
• Metric Unit Conversion
• Customary Unit Conversion
• Temperature
• More Measurement Worksheets
• Writing Checks
• Profit and Loss
• Simple Interest
• Compound Interest
• Tally Marks
• Mean, Median, Mode, Range
• Mean Absolute Deviation
• Stem-and-leaf Plot
• Box-and-whisker Plot
• Permutation and Combination
• Probability
• Venn Diagram
• More Statistics Worksheets
• Shapes - 2D
• Shapes - 3D
• Lines, Rays and Line Segments
• Points, Lines and Planes
• Transformation
• Quadrilateral
• Ordered Pairs
• Midpoint Formula
• Distance Formula
• Parallel, Perpendicular and Intersecting Lines
• Scale Factor
• Surface Area
• Pythagorean Theorem
• More Geometry Worksheets
• Converting between Fractions and Decimals
• Significant Figures
• Convert between Fractions, Decimals, and Percents
• Proportions
• Direct and Inverse Variation
• Order of Operations
• Squaring Numbers
• Square Roots
• Scientific Notations
• Speed, Distance, and Time
• Absolute Value
• More Pre-Algebra Worksheets
• Translating Algebraic Phrases
• Evaluating Algebraic Expressions
• Simplifying Algebraic Expressions
• Algebraic Identities
• Quadratic Equations
• Systems of Equations
• Polynomials
• Inequalities
• Sequence and Series
• Complex Numbers
• More Algebra Worksheets
• Trigonometry
• Math Workbooks
• English Language Arts
• Summer Review Packets
• Social Studies
• Holidays and Events
• Worksheets >
• Algebra >
• Equations >

Absolute Value Equation and Function Worksheets

Explore this ensemble of printable absolute value equations and functions worksheets to hone the skills of high school students in evaluating absolute functions with input and output table, evaluating absolute value expressions, solving absolute value equations and graphing functions. Give a head-start to your practice with the free worksheets here.

Solving Absolute Value Expression

Input and Output Tables

Using the input values and the rule, find out the output values of the absolute value function.

Input and Output Tables: Easy

Input and Output Tables: Moderate

Input and Output Tables: Difficult

• Download All

Evaluate Absolute Value Expression

Find the value of each absolute value expression for the particular value of x.

Evaluate Absolute Value: Easy

Evaluate Absolute Value: Moderate

Evaluate Absolute Value: Difficult

Solving Absolute Value Equation

Existence of the Solution:

Supply this set of printable handouts to high school students and knock their skills into shape! They need to check if a given equation has a solution or not.

Sheet 1 | Sheet 2 | Grab 'em All

Type 1: | x | = a

Solve the absolute value equation and find the solutions. The question either has two solutions or no solution.

Type 2: |x + a| = b

Set x + a = b and x + a = -b, and solve the absolute value equations.

Easy: Sheet 1 | Sheet 2 | Grab 'em All

Moderate: Sheet 1 | Sheet 2 | Grab 'em All

Type 3: a |bx + c| = d

Isolate the absolute value expression and then solve.

Type 4: a |bx + c| + d = e

Isolate the absolute value expression from the constant term and solve for x.

Type 5: |ax + b| = |cx + d|

All you need to do in this pdf practice set is this: find the break points, set the interval, solve for x and check for an extraneous solution.

Graphing Absolute Value Function

Identify the Absolute Value Function

The graph of the absolute value functions includes vertical shift, horizontal shift and reflection. Identify the absolute value function in each graph.

Difficult: Sheet 1 | Sheet 2 | Grab 'em All

Each question in this set of pdf worksheets contains a function and a grid displaying the x and y coordinates below. Identify the transformations involved in the function and then graph.

Vertical / Horizontal shift:

Graphing Functions: Easy 1

Graphing Functions: Easy 2 | Grab 'em All

Vertical shift + Horizontal shift:

Graphing Functions: Moderate 1

Graphing Functions: Moderate 2 | Grab 'em All

Vertical shift + Horizontal shift + Reflection:

Graphing Functions: Difficult 1

Graphing Functions: Difficult 2 | Grab 'em All

Related Worksheets

» One-step Equation

» Two-step Equation

» Multi-step Equation

» Quadratic Equation

» Equations

Become a Member

Membership Information

Privacy Policy

What's New?

Printing Help

Testimonial

Copyright © 2024 - Math Worksheets 4 Kids

This is a members-only feature!

Graphing Absolute Value Functions

How to graph absolute value functions.

This lesson is about graphing an absolute value function when the expression inside the absolute value symbol is linear. It is linear if the variable “[latex]x[/latex]” has a power of [latex]1[/latex]. The graph of absolute value function has a shape of “V” or inverted “V”.

Absolute Value Function in Equation Form

The general form of an absolute value function that is linear is

• where the vertex (low or high point) is located at 

• the vertical line

divides the graph into two equal halves

KEY POINTS TO REMEMBER:

• The specific approach that we are going to use is to find the “right amount of points” that can be plotted on the [latex]xy[/latex]-axis so that we have a good approximation on how the graph of the absolute value function looks likes.
• Don’t fall into the common trap to always use [latex]x = 0[/latex] as the middle value of the [latex]x[/latex] coordinates. This may work at times, but this idea is not so reliable.
• I suggest using the [latex]x[/latex]-coordinate of the vertex, [latex]\left( {{{ – b} \over m},\,c} \right)[/latex], which is [latex]x = {{ – b} \over m}[/latex] as the middle value of all [latex]x[/latex]-values in the table.
• In other words, once you have determined what the middle value for [latex]x[/latex] is, then go ahead and pick any arbitrary [latex]x[/latex] values to the left and right of it as long as it is properly spaced out.

Again, the best way to illustrate this simple idea is through the use of examples!

Examples of Graphing Absolute Value Functions

Example 1: Graph the absolute value function below using the table of values.

This is the most basic form of an absolute value function. If you see that the only expression inside the absolute value symbol is just “[latex]x[/latex]”, assume that the vertex of the graph will occur when [latex]x = 0[/latex].

Or, if you want to use the formula above to find the vertex, rewrite the given function in the standard form so that you can identify the values of [latex]m[/latex], [latex]b[/latex], and [latex]c[/latex].

The vertex is calculated as

Since [latex]x = 0[/latex], this becomes the center values of all [latex]x[/latex]’s. Now we can pick some numbers to the left and to the right of zero. I would suggest using equal amount of numbers that are of the same increment.

Pick [latex]-3[/latex], [latex]-2[/latex], [latex]-1[/latex] to the left of zero, and [latex]+1[/latex], [latex]+2[/latex], [latex]+3[/latex] to its right. Evaluate all values of [latex]x[/latex] into the function [latex]y = \left| x \right|[/latex] to get the corresponding [latex]y[/latex]-values.

Notice that the graph has a low point determined by the middle [latex]x[/latex]-value which is the [latex]x[/latex]-coordinate of the vertex itself, i.e. ( 0 ,0).

Example 2: Graph the absolute value function below using the table of values.

The first step is to find the [latex]x[/latex]-coordinate of the vertex which will serve as the center point in the table of values of [latex]x[/latex].

Rewrite [latex]y = \left| {x – 2} \right|\,\,\,[/latex]as [latex]y = \left| {1x + \left( { – 2} \right)} \right| + 0[/latex] where [latex] m = 1[/latex], [latex]b = -2[/latex], and [latex]c = 0[/latex]. We calculate the vertex as…

The [latex]x[/latex]-coordinate of the vertex will be the center value of all [latex]x[/latex]’s on the table. We generated the rest of the [latex]x[/latex] by finding three numbers to the left and right of the middle value of [latex]2[/latex] with an increment of [latex]1[/latex]. You may use an increment of [latex]2[/latex], and trust me, the graph will be the same.

Plot the points on the [latex]xy[/latex]-plane and connect the dots with a straight edge. If you get it right, you should have something similar below. As you can see, the low point of the graph is the vertex located at ([latex]2[/latex],[latex]0[/latex]).

Example 3: Graph the absolute value function below using the table of values.

I hope you start realizing that the first step is to always express the given absolute value function in standard form. This allows us to identify the correct values of [latex]m[/latex], [latex]b[/latex], and [latex]c[/latex] which we will use to substitute into the formula.

Clearly, the required values are [latex]m = 2[/latex], [latex]b = 6[/latex], and [latex]c = -4[/latex].

Then we calculate the vertex as follows;

Our table of values will have a center value of [latex]x = – 3[/latex]. Generate 3 numbers both to the left and right of [latex]x = – 3[/latex] with an increment of [latex]1[/latex]. Then evaluate each value of [latex]x[/latex] into the function [latex]y = \left| {\,2x + 6\,} \right| – 4\,[/latex] to get the corresponding values of [latex]y[/latex] in the table.

Your table should look something like this

Plot the points on the Cartesian plane and connect them using a straight edge such as a ruler.

Example 4: Graph the absolute value function below using the table of values.

This is an example of an absolute value function whose graph is an inverted “V”. This happens because the coefficient of the absolute value symbol is negative, that is, [latex] – 1[/latex].

Let’s rewrite this in the standard form.

That means [latex]m = 1[/latex], [latex]b = -3[/latex], and [latex]c = -2[/latex].

Solving the vertex of the function,

The table of values will have a center value of [latex]x = 3[/latex].

Plotting the points in the [latex]xy[/latex]-axis,

You may also be interested in these related math lessons or tutorials:

Solving Absolute Value Equations Solving Absolute Value Inequalities

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.9: Absolute Value Functions

• Last updated
• Save as PDF
• Page ID 59844

• David Arnold
• College of the Redwoods
• 1.9.1: Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is extremely important to have a good grasp of the concept of a piecewise-defined function.
• 1.9.2: Absolute Value The absolute value of a number is a measure of its magnitude, sans (without) its sign.
• 1.9.3: Absolute Value Equations
• 1.9.4: Absolute Value Inequalities

Thumbnail: The graph of the absolute value function for real numbers. (CC BY-SA 3.0; Qef and Ævar Arnfjörð Bjarmason via Wikipedia).