Learning Objectives
- Graph an absolute value function.
- Solve an absolute value equation.
- Solve an absolute value inequality.
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 investigate absolute value functions.
Understanding Absolute Value
Recall that in its basic form 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.
Absolute Value Function
The absolute value function can be defined as a piecewise function
Example 1
Determine a Number within a Prescribed Distance
Describe all values within or including a distance of 4 from the number 5.
Analysis
Note that
So is equivalent to
However, mathematicians generally prefer absolute value notation.
Describe all values within a distance of 3 from the number 2.
Example 2
Resistance of a Resistor
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 or
Suppose we have a resistor rated at 680 ohms, Use the absolute value function to express the range of possible values of the actual resistance.
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 3.
Figure 4 shows the graph of The graph of 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 for this transformed function.
Example 3
Writing an Equation for an Absolute Value Function
Write an equation for the function graphed in Figure 5.
Analysis
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.
Q&A
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 and
Now substituting in the point (1, 2)
Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.
Q&A
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 8).
Solving an Absolute Value Equation
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 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 and an equation of the form with will have solutions when or If the equation has no solution.
How To
Given the formula for an absolute value function, find the horizontal intercepts of its graph.
- Isolate the absolute value term.
- Use to write or assuming
- Solve for
Example 4
Finding the Zeros of an Absolute Value Function
For the function , find the values of such that .
For the function find the values of such that
Q&A
Should we always expect two answers when solving
No. We may find one, two, or even no answers. For example, there is no solution to
How To
Given an absolute value equation, solve it.
- Isolate the absolute value term.
- Use to write or
- Solve for
Example 5
Solving an Absolute Value Equation
Solve
Q&A
Find where the graph of the function intersects the horizontal and vertical axes.
Solving an Absolute Value Inequality
Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form
where an expression (and possibly but not usually ) depends on a variable Solving the inequality means finding the set of all that satisfy the inequality. Usually this set will be an interval or the union of two intervals.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.
For example, we know that all numbers within 200 units of 0 may be expressed as
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values such that the distance between and 600 is less than 200. We represent the distance between and 600 as
This means our returns would be between $400 and $800.
Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.
How To
Given an absolute value inequality of the form for real numbers and where is positive, solve the absolute value inequality algebraically.
- Find boundary points by solving
- Test intervals created by the boundary points to determine where
- Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.
Example 6
Solving an Absolute Value Inequality
Solve
Analysis
For absolute value inequalities,
The or symbol may be replaced by
So, for this example, we could use this alternative approach.
Solve
How To
Given an absolute value function, solve for the set of inputs where the output is positive (or negative).
- Set the function equal to zero, and solve for the boundary points of the solution set.
- Use test points or a graph to determine where the function’s output is positive or negative.
Example 7
Using a Graphical Approach to Solve Absolute Value Inequalities
Given the function determine the values for which the function values are negative.
Solve
Media
Access these online resources for additional instruction and practice with absolute value.
1.6 Section Exercises
Verbal
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?
Algebraic
Describe all numbers that are at a distance of 4 from the number 8. Express this using absolute value notation.
Describe all numbers that are at a distance of from the number −4. Express this using absolute value notation.
Describe the situation in which the distance that point is from 10 is at least 15 units. Express this using absolute value notation.
Find all function values such that the distance from to the value 8 is less than 0.03 units. Express this using absolute value notation.
For the following exercises, solve the equations below and express the answer using set notation.
For the following exercises, find the x- and y-intercepts of the graphs of each function.
For the following exercises, solve each inequality and write the solution in interval notation.
Graphical
For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.
For the following exercises, graph the given functions by hand.
Technology
Use a graphing utility to graph on the viewing window Identify the corresponding range. Show the graph.
Use a graphing utility to graph on the viewing window Identify the corresponding range. Show the graph.
For the following exercises, graph each function using a graphing utility. Specify the viewing window.
Extensions
For the following exercises, solve the inequality.
If possible, find all values of such that there are no intercepts for
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 represents the distance from city B to city A, express this using absolute value notation.
The true proportion 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 for the score.
A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using 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 inches, express the tolerance using absolute value notation.