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Absolute Value Functions

Graph an absolute value function.

Absolute Value Functions

  1. Solve absolute value equations (IA 2.7.1)
  2. Identify graphs of absolute value functions (IA 3.6.2)

Recall that in its basic form, 𝑓(𝑥)=|𝑥|, the absolute value function is one of our toolkit functions. The absolute value function is often thought of as providing the distance the number is from zero on a number line. Numerically, for whatever the input value is, the output is the magnitude of this value.

The absolute value function can be defined as a piecewise function

\(f(x)=|x|=-x\) , when \(x<0\) or \(x\) , when \(x\ge 0\)

Solve absolute value equations

Try it.

ⓐ Solve for x: |x|=5

Solution

Since the absolute value of a number is its distance from 0 on the number line. Notice that both 5 and –5 are 5 units from 0 on the number line.

If |x|=5
x=5, or x=–5

Try it.

ⓑ Solve for x: |2x–5|=3

Solution

In this case the number represented by 2x–5 is 3 units from zero on the number line. So 2x–5 could either equal 3 or –3.

\(2x-5=3\ 2x=8\ x=4\\)

\(2x-5=-3\ 2x=2\ x=1\\)

Solve absolute value equations.

Try it.

Solve for x: \(|x|=0\)

Try it.

Solve for x: \(|x|=2\)

Try it.

Solve for x: \(|x-2|=6\)

Try it.

Solve for x: \(|2x+1|=7\)

Try it.

Solve for t: \(|3t-1|=-2\)

Try it.

Solve for z: \(|2z-3|-4=1\)

Try it.

Solve for x: \(-2|x-3|+8=-4\)

Absolute value functions have a “V” shaped graph. If scanning this function from left to right the corner is the point where the graph changes direction.

\(f(x)=|x|=-x\) , when \(x<0\) or \(x\) , when \(x\ge 0\)

Identify and graph absolute value functions. Graph each of the following functions. Label at least one point on your graph.

Try it.

\(y=|x|-2\)

Try it.

\(y=|2x|+3\)

Try it.

\(y=|x-4|+3\)

Try it.

\(y=-|x+6|-4\)

Try it.

\(y=-|x-2|\)

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\(y=3|x+1|+4\)

Try it.

Find the domain and range of the following absolute value function.

Understanding Absolute Value

Recall that in its basic form \(f(x)=|x|,\) 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.

Example

Try it.

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 \(\text{\pm 1\%,}\ \pm \text{5\%,}\) or \(\pm \text{10\%}\text{.}\)

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

Solution

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 \(R\) in ohms,

\[|R-680|\le 34\]

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 .

shows the graph of \(y=2|x-3|+4.\) The graph of \(y=|x|\) 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 \((3,4)\) for this transformed function.

Example

Try it.

Write an equation for the function graphed in .

Solution

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 .

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 .

From this information we can write the equation

\[\begin{array}{llll}f(x) & = & 2|x-3|-2, & \ \text{treating the stretch as }a\ \text{vertical stretch,or} \\ f(x) & = & |2(x-3)|-2, & \ \text{treating the stretch as }a\ \text{horizontal compression}.\end{array}\]

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 \(8=|2x-6|,\) 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.

\[\begin{array}{lllllll}2x-6 & = & 8 & \ \text{or}\ & 2x-6 & = & -8 \\ 2x & = & 14 & & 2x & = & -2 \\ x & = & 7 & & x & = & -1\end{array}\]

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,

\[\begin{array}{l}|x|=4, \\ |2x-1|=3,\text{or} \\ |5x+2|-4=9\end{array}\]
Example

Try it.

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

Solution\[\begin{array}{llllllll}0 & = & |4x+1|-7 & & & & & \text{Substitute 0 for }f(x). \\ 7 & = & |4x+1| & & & & & \text{Isolate the absolute value on one side of the equation}. \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ 7 & = & 4x+1 & \text{or} & \ -7 & = & 4x+1 & \text{Break into two separate equations and solve}. \\ 6 & = & 4x & & -8 & = & 4x & \\ & & & & & & & \\ x & = & \frac{6}{4}=1.5 & & x & = & \frac{-8}{4}=-2 & \end{array}\]

The function outputs 0 when \(x=\frac{3}{2}\) or \(x=-2.\) See .

Absolute Value Functions

Key Concepts

  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See .
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See .
  • 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 .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. ⓐ Solve for x: |x|=5

    Kusonyeza yankho

    Since the absolute value of a number is its distance from 0 on the number line. Notice that both 5 and –5 are 5 units from 0 on the number line.

    If |x|=5
    x=5, or x=–5

  2. ⓑ Solve for x: |2x–5|=3

    Kusonyeza yankho

    In this case the number represented by 2x–5 is 3 units from zero on the number line. So 2x–5 could either equal 3 or –3.

    \(2x-5=3\ 2x=8\ x=4\\)

    \(2x-5=-3\ 2x=2\ x=1\\)

  3. Solve for x: \(|x|=0\)

  4. Solve for x: \(|x|=2\)

  5. Solve for x: \(|x-2|=6\)

  6. Solve for x: \(|2x+1|=7\)

  7. Solve for t: \(|3t-1|=-2\)

  8. Solve for z: \(|2z-3|-4=1\)

  9. Solve for x: \(-2|x-3|+8=-4\)

  10. \(y=|x|-2\)

  11. \(y=|2x|+3\)

  12. \(y=|x-4|+3\)

  13. \(y=-|x+6|-4\)

  14. \(y=-|x-2|\)

  15. \(y=3|x+1|+4\)

  16. Find the domain and range of the following absolute value function.

  17. 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 \(\text{\pm 1\%,}\ \pm \text{5\%,}\) or \(\pm \text{10\%}\text{.}\)

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

    Kusonyeza yankho

    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 \(R\) in ohms,

    \[|R-680|\le 34\]

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

    Kusonyeza yankho

    using the variable \(p\) for passing, \(|p-80|\le 20\)

  19. Write an equation for the function graphed in .

    Kusonyeza yankho

    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 .

    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 .

    From this information we can write the equation

    \[\begin{array}{llll}f(x) & = & 2|x-3|-2, & \ \text{treating the stretch as }a\ \text{vertical stretch,or} \\ f(x) & = & |2(x-3)|-2, & \ \text{treating the stretch as }a\ \text{horizontal compression}.\end{array}\]

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

    Kusonyeza yankho

    \(f(x)=-|x+2|+3\)

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

    Kusonyeza yankho
    \[\begin{array}{llllllll}0 & = & |4x+1|-7 & & & & & \text{Substitute 0 for }f(x). \\ 7 & = & |4x+1| & & & & & \text{Isolate the absolute value on one side of the equation}. \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ 7 & = & 4x+1 & \text{or} & \ -7 & = & 4x+1 & \text{Break into two separate equations and solve}. \\ 6 & = & 4x & & -8 & = & 4x & \\ & & & & & & & \\ x & = & \frac{6}{4}=1.5 & & x & = & \frac{-8}{4}=-2 & \end{array}\]

    The function outputs 0 when \(x=\frac{3}{2}\) or \(x=-2.\) See .

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

    Kusonyeza yankho

    \(x=-1\) or \(x=2\)

  23. How do you solve an absolute value equation?

    Kusonyeza yankho

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

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

  25. 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?

    Kusonyeza yankho

    The graph of the absolute value function does not cross the \(x\) -axis, so the graph is either completely above or completely below the \(x\) -axis.

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

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

    Kusonyeza yankho

    The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4.

  28. Describe all numbers \(x\) that are at a distance of \(\frac{1}{2}\) from the number −4. Express this set of numbers using absolute value notation.

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

    Kusonyeza yankho

    ∣ x − 10 ∣ ≥ 15

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

  31. \(f(x)=4|x-3|+4\)

    Kusonyeza yankho

    There are no x-intercepts.

  32. \(f(x)=-3|x-2|-1\)

  33. \(f(x)=-2|x+1|+6\)

    Kusonyeza yankho

    (−4, 0) and (2, 0)

  34. \(f(x)=-5|x+2|+15\)

  35. \(f(x)=2|x-1|-6\)

    Kusonyeza yankho

    \((0,-4),(4,0),(-2,0)\)

  36. \(f(x)=|-2x+1|-13\)

  37. \(f(x)=-|x-9|+16\)

    Kusonyeza yankho

    \((0,7),(25,0),(-7,0)\)

  38. \(y=|x-1|\)

  39. \(y=|x+1|\)

  40. \(y=|x|+1\)

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Absolute Value Functions

  1. Graph an absolute value function.
  2. Solve an absolute value equation.
  3. Solve absolute value equations (IA 2.7.1)
  4. Identify graphs of absolute value functions (IA 3.6.2)
  5. Isolate the absolute value term.
  6. Use
  7. Solve for
  8. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Sankhani wanu

Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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