Lesson 14

Absolute Value Functions (Part 2)

Let’s investigate distance as a function.

14.1: Temperature in Toronto

Toronto is a city at the border of the United States and Canada, just north of Buffalo, New York. Here are twelve guesses of the average temperature of Toronto, in degrees Celsius, in February 2017. 

  • 5
  • 2
  • -5
  • 3
  • 0
  • -1
  • 1.5
  • 4
  • -2.5
  • 6
  • 4
  • -0.5
  1. The actual average temperature of Toronto in February 2017 is 0 degree Celsius.

    Use this information to sketch a scatter plot representing the guesses, \(x\), and the corresponding absolute guessing errors, \(y\).

    horizontal axis, guess. scale -8 to 8, by 1's. vertical axis, absolute guessing error. scale 0 to 8, by 1's. 
  2. What rule can you write to find the output given the input?

14.2: The Distance Function

The function \(A\) gives the distance of \(x\) from 0 on the number line.

  1. Complete the table and sketch a graph of function \(A\).
    \(x\) \(A(x)\)
    8  
      5.6
    \(\pi\)  
    \(\frac12\)  
      1
    0  
    \(\text-\frac12\)  
    -1  
    -5.6  
      8
    Coordinate plane, x and y axis, scale -8 to 8, by 2's.
  2. Andre and Elena are trying to write a rule for this function.

    • Andre writes: \(A(x)=\begin{cases} x,& x \geq 0 \\ \text-x, & x < 0 \\ \end{cases}\)
    • Elena writes: \(A(x) = |x|\)

    Explain why both equations correctly represent the function \(A\).

14.3: Moving Graphs Around

Graph the functions \( f(x)=|x+a|\) and \(g(x)=|x|+b\). Experiment with different values of \(a\) and \(b\) and observe what happens to the graphs.

  1. How does changing \(a\) change the graph?
  2. How does changing \(b\) change the graph?


  1. Look for the minimum of each graph. Each point represents the least value of the function.
    1. For function \(f\), what value of \(x\) gives the least output value?
    2. For function \(g\), what value of \(x\) gives the least output value?
  2. Another function is defined by \(m(x)=|x+11.5|\).
    1. What value of \(x\) produces the least output of function \(m\)? Be prepared to explain how you know.
    2. Describe the graph of \(m\).

14.4: More Moving Graphs Around

Here are five equations and four graphs.

  • Equation 1: \(y=|x-3|\)
  • Equation 2: \(y=|x-9|+3\)
  • Equation 3: \(y=|x|-6\)
  • Equation 4: \(y=|x+3|\)
  • Equation 5: \(y=|x+3|-6\)

A

Graph of absolute value function with vertex at 0 comma -6.

B

Graph of absolute value function with vertex at 3 comma 0.

C

Graph of absolute value function with vertex at -3 comma -6

D

graph of absolute value function with vertex at 9 comma 3.

E

x and y axes, scale -12 to 12, by 3's. 

  1. Match each equation with a graph that represents it. One equation has no match.
  2. For the equation without a match, sketch a graph on the blank coordinate plane.
  3. Use graphing technology to check your matches and your graph. Revise your matches and graphs as needed.

Summary

In a guessing game, each guess can be seen as an input of a function and each absolute guessing error as an output. Because absolute guessing error tells us how far a guess is from a target number, the output is distance.

Suppose the target number is 0.

  • We can find the distance of a guess, \(x\), from 0 by calculating \(x-0\). Because distance cannot be negative, what we want to find is \(|x-0|\), or simply \(|x|\).
  • If function \(f\) gives the distance of \(x\) from 0, we can define it with the equation:

\(f(x)=|x|\)

Function \(f\) is the absolute value function. It gives the distance of \(x\) from 0 by finding the absolute value of \(x\).

The graph of function \(f\) is a V shape with the two lines converging at \((0,0)\).

We call this point the vertex of the graph. It is the point where a graph changes direction, from going down to going up, or the other way around.

We can also think of a function like \(f\) as a piecewise function because different rules apply when \(x\) is less than 0 and when it is greater than 0.

Graph of absolute value function with vertex at origin.

Suppose we want to find the distance between \(x\) and 4.

  • We can find the difference between \(x\) and 4 by calculating \(x-4\). Distance cannot be negative, so what we want is the absolute value of that difference: \(|x-4|\).
  • If function \(p\) gives the distance of \(x\) from 4, we can define it with the equation:

\(p(x)= |x - 4|\)

Graph of absolute value function p with vertex at 4 comma 0.

Now suppose we want to find the distance between \(x\) and -4.

  • We can find the difference of \(x\) and -4 by calculating \(x-(\text-4)\), which is equal to \(x+4\). Distance cannot be negative, so let's find the absolute value: \(|x+4|\).
  • If function \(q\) gives the distance of \(x\) from -4, we can define it with the equation:

\(q(x)= |x + 4|\)

Graph of absolute value function q with vertex at -4 comma 0.

Notice that the graphs of \(p\) and \(g\) are like that of \(f\), but they have shifted horizontally.

Glossary Entries

  • absolute value

    The absolute value of a number is its distance from 0 on the number line. 

  • domain

    The domain of a function is the set of all of its possible input values.

  • piecewise function

    A piecewise function is a function defined using different expressions for different intervals in its domain.

  • range

    The range of a function is the set of all of its possible output values.