Absolute Value FunctionsMathematics and Statistics
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 formthe 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.
The absolute value function can be defined as a piecewise function
Describe all valueswithin or including a distance of 4 from the number 5.
We want the distance betweenand 5 to be less than or equal to 4. We can draw a number line, such as the one in [link], to represent the condition to be satisfied.
The distance fromto 5 can be represented using the absolute value asWe want the values ofthat satisfy the condition
Describe all valueswithin a distance of 3 from the number 2.
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, oftenor
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.
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 resistancein 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 variablefor passing,
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 [link].
[link] shows the graph ofThe graph ofhas 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 atfor this transformed function.
Write an equation for the function graphed in [link].
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 [link].
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 [link].
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 forand
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.
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 [link]).
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 aswe 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,
For real numbersandan equation of the formwithwill have solutions whenorIfthe equationhas no solution.
Given the formula for an absolute value function, find the horizontal intercepts of its graph.
- Isolate the absolute value term.
- Useto writeorassuming
- Solve for
For the function, find the values of such that .
The function outputs 0 whenor See [link].
For the functionfind the values ofsuch that
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
Given an absolute value equation, solve it.
- Isolate the absolute value term.
- Useto writeor
- Solve for
Isolating the absolute value on one side of the equation gives the following.
The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.
In [link], ifandwere graphed on the same set of axes, would the graphs intersect?
No. The graphs ofandwould not intersect, as shown in [link]. This confirms, graphically, that the equationhas no solution.
Find where the graph of the functionintersects the horizontal and vertical axes.
so the graph intersects the vertical axis atwhenandso the graph intersects the horizontal axis atand
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 variableSolving the inequality means finding the set of allthat 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 betweenand 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.
Given an absolute value inequality of the formfor real numbersandwhereis 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.
With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find whereWe do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve
After determining that the absolute value is equal to 4 atandwe know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:
To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in [link].
Becauseis the only interval in which the output at the test value is less than 4, we can conclude that the solution toisor
To use a graph, we can sketch the functionTo help us see where the outputs are 4, the linecould also be sketched as in [link].
We can see the following:
- The output values of the absolute value are equal to 4 atand
- The graph ofis below the graph ofonThis means the output values ofare less than the output values of
- The absolute value is less than or equal to 4 between these two points, whenIn interval notation, this would be the interval
For absolute value inequalities,
Theorsymbol may be replaced by
So, for this example, we could use this alternative approach.
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.
Given the function determine the values for which the function values are negative.
We are trying to determine wherewhich is whenWe begin by isolating the absolute value.
Next we solve for the equality
Now, we can examine the graph ofto observe where the output is negative. We will observe where the branches are below the x-axis. Notice that it is not even important exactly what the graph looks like, as long as we know that it crosses the horizontal axis atandand that the graph has been reflected vertically. See [link].
We observe that the graph of the function is below the x-axis left ofand right ofThis means the function values are negative to the left of the first horizontal intercept atand negative to the right of the second intercept atThis gives us the solution to the inequality.
In interval notation, this would be
orin interval notation, this would be
Access these online resources for additional instruction and practice with absolute value.
- The absolute value function is commonly used to measure distances between points. See [link].
- Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See [link].
- The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See [link].
- 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. See [link].
- An absolute value equation may have one solution, two solutions, or no solutions. See [link].
- An absolute value inequality is similar to an absolute value equation but takes the form It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See [link].
- Absolute value inequalities can also be solved graphically. See [link].
How do you solve an absolute value equation?
Isolate the absolute value term so that the equation is of the formForm one equation by setting the expression inside the absolute value symbol,equal to the expression on the other side of the equation,Form a second equation by settingequal to the opposite of the expression on the other side of the equation,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-axis, so the graph is either completely above or completely below the-axis.
How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?
How do you solve an absolute value inequality algebraically?
First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.
Describe all numbersthat are at a distance of 4 from the number 8. Express this using absolute value notation.
Describe all numbersthat are at a distance offrom the number −4. Express this using absolute value notation.
Describe the situation in which the distance that pointis from 10 is at least 15 units. Express this using absolute value notation.
Find all function valuessuch that the distance fromto 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.
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.
Use a graphing utility to graph on the viewing window Identify the corresponding range. Show the graph.
Use a graphing utility to graphon the viewing windowIdentify the corresponding range. Show the graph.
For the following exercises, graph each function using a graphing utility. Specify the viewing window.
For the following exercises, solve the inequality.
If possible, find all values of such that there are no intercepts for
If possible, find all values ofsuch that there are no -intercepts for
There is no solution forthat will keep the function from having a-intercept. The absolute value function always crosses the -intercept when
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 andrepresents the distance from city B to city A, express this using absolute value notation.
The true proportionof 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 variablefor the score.
A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Usingas 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 isinches, express the tolerance using absolute value notation.
Sois equivalent to
However, mathematicians generally prefer absolute value notation.