ORCCA Introduction to Absolute Value Functions

Section 12.1 Introduction to Absolute Value Functions

Subsection 12.1.1 Definition of Absolute Value

Recall that in Section 1.4, we defined the absolute value of a number to be the distance between that number and \(0\) on a number line. Also recall that this causes the output of the absolute value function to never be a negative number since we are under the presumption that “distance” is always positive (or zero).

Example 12.1.1.

Since the number \(5\) is \(5\) units from \(0\text{,}\) then \(\abs{5}=5\text{.}\)
Since the number \(-3\) is \(3\) units from \(0\text{,}\) then \(\abs{-3}=3\text{.}\)

Example 12.1.2.

Yonas takes a \(5\)-block walk north from his home to a food cart. After enjoying dinner, he then walks \(9\) blocks south of the food cart to his favorite movie theater.

How many blocks has Yonas walked in total when he reaches the theater?
How many blocks is Yonas from home when he reaches the theater?

Explanation

Since we only care about total distance, we can ignore the “signs” on the distances walked (either north or south) and simply add the two values together. Mathematically, if we think of north as positive values and south as having negative values, this situation is the same as

\begin{align*} \abs{5}+\abs{-9}\amp=5+9\\ \amp=14 \end{align*}

Yonas walked a total of \(14\) blocks when he reached the theater.
When he reaches the theater, Yonas's actual position could be thought of as \(5+(-9)\text{.}\) But the actual distance from the theater to his home is better thought of as:

\begin{align*} \abs{5+(-9)}\amp=\abs{-4}\\ \amp=4 \end{align*}

Yonas was \(4\) blocks from home when he reached the theater.

Subsection 12.1.2 Evaluating Absolute Value Functions

The formula \(f(x)=\abs{x}\) does satisfy the requirements for \(f\) to be a function because no matter what number you put in for \(x\text{,}\) there is only one measured distance from \(0\) to that value \(x\text{.}\)

Example 12.1.3.

Let \(f(x)=\abs{x}\) and \(g(x)=\abs{2x-5}\text{.}\) Evaluate the following expressions.

\(\displaystyle f(34)\)
\(\displaystyle f(-63)\)
\(\displaystyle f(0)\)
\(\displaystyle g(13)\)
\(\displaystyle g(1)\)

Explanation

\(\displaystyle \begin{aligned}[t] f(\highlight{34})\amp=\abs{\highlight{34}}\\ \amp=34 \end{aligned}\)
\(\displaystyle \begin{aligned}[t] f(\highlight{-63})\amp=\abs{\highlight{-63}}\\ \amp=63 \end{aligned}\)
\(\displaystyle \begin{aligned}[t] f(\highlight{0})\amp=\abs{\highlight{0}}\\ \amp=0 \end{aligned}\)
\(\displaystyle \begin{aligned}[t] g(\highlight{13})\amp=\abs{2\cdot\highlight{13}-5}\\ \amp=\abs{21}\\ \amp=21 \end{aligned}\)
\(\displaystyle \begin{aligned}[t] g(\highlight{1})\amp=\abs{2\cdot\highlight{1}-5}\\ \amp=\abs{-3}\\ \amp=3 \end{aligned}\)

Checkpoint 12.1.4.

Subsection 12.1.3 Graphs of Absolute Value Functions

Absolute value functions have generally the same shape. They are usually described as “V” -shaped graphs and the tip of the “V” is called the vertex. A few graphs of various absolute value functions are shown in Figure 12.1.5. In general, the domain of an absolute value function (where there is a polynomial inside the absolute value) is \((-\infty,\infty)\text{.}\)

A Cartesian graph with the graph of y=abs(x), which looks like a V with the vertex of the V at the origin. The right side extends as a line with slope 1 and the left side extends with slope -1. — (a) \(y=\abs{x}\)

A Cartesian graph with the graph of y=-abs(x+2), which looks like an upside down V with the vertex of the V at the point (-2,0). The right side extends as a line with slope -1 and the left side extends with slope 1. — (a) \(y=\abs{x}\)

Example 12.1.6.

Let \(h(x)=-2\abs{x-3}+5\text{.}\) Create a table of ordered pairs with \(x\)-values from \(-3\) to \(3\text{,}\) using an increment of \(1\text{.}\) Then sketch a graph of \(y=h(x)\text{.}\) State the domain and range of \(h\text{.}\)

Explanation

Table 12.1.7. Table for \(y=h(x)\text{.}\)

\(x\)	\(y\)
\(-3\)	\(-7\)
\(-2\)	\(-5\)
\(-1\)	\(-3\)
\(0\)	\(-1\)
\(1\)	\(1\)
\(2\)	\(3\)
\(3\)	\(5\)

A Cartesian graph with the graph of y=-2*abs(x-3)+5, which looks like an upside down V with the vertex of the V at the point (3,5). The right side extends as a line with slope -2 and the left side extends with slope 2. — Figure 12.1.8. Graph of \(y=h(x)\)

The graph indicates that the domain is \((\infty,\infty)\) as it goes to the right and left indefinitely. The range is \((-\infty,5]\text{.}\)

Subsection 12.1.4 Applications Involving Absolute Values

Absolute values are quite useful as models in a variety of real world applications. One example is the path of a billiards (pool) ball: when the ball bounces off one of the side rails, its path is mirrored and creates a “V” shape. The game gets more complicated when more than the rail is hit, but the fundamental mathematics doesn't change: absolute values model the bounces each time.

Here are some more examples. The first one we'll explore involves light reflecting off of a mirror.

Example 12.1.9.

When light reflects off of a mirror, the path it takes is in the shape of an absolute value graph. Khenbish was playing with a laser pointer in his bedroom mirror. He set up the laser pointer on his windowsill and the light hit the center of the mirror and reflected onto the corner of his room. He declared that the laser pointer is sitting at the origin, and \(x\) should stand for the horizontal distance from the left wall to the light beam. Shown is a birds-eye view of the situation.

a Cartesian graph to show the top view of the room with a line starting at the origin and traveling upward, bouncing off the mirror and hitting the bottom corner of the room, making an upside down V-shaped graph — Figure 12.1.10. Birds-Eye View of Khenbish's Room with Laser

After a little bit of work, Khenbish was able to come up with a formula for the light's path:

\begin{equation*} p(x)=5-\frac{5}{4}\abs{x-4} \end{equation*}

where \(p(x)\) stands for the position, in ft, above (for positive values) or below (for negatives) the center line through his room that represents the \(x\)-axis, where \(x\) is also measured in ft. Use a graph of this formula to answer the following questions.

Khenbish's room is 10 ft wide according to Figure 12.1.10 (in the vertical direction in the figure). What is the room's length (in the horizontal direction in the figure)?
How far along the wall is the mirror centered?
If you stood 9 ft from the left wall, how far above or below the room's center line (\(x\)-axis) should you stand to have the laser pointer hit you?

Explanation

the previous graph with a grid added so it is clear that the mirror is at the point (4,5) and the point where the light hits the corner is (12,-5) — Figure 12.1.11. Detailed Birds-Eye View of Khenbish's Room

To find the room's length, first note that since the laser hits the corner of the room, the \(x\) coordinate of the lasers position would tell us the room's width. According to the detailed graph, the \(x\)-coordinate when \(y=-5\) is \(12\text{.}\) So the room must be \(12\) feet wide.
The mirror is centered exactly where the laser hits the wall. This is the vertex of the absolute value graph which, according to the graph, is at the point \((4,5)\text{.}\) This tells us that the mirror is centered \(4\) feet from the left wall.
If you are standing 9 feet from the left wall, the laser's position will be a bit more than one foot behind the rooms center line, by the diagram. Here is how to do this problem algebraically.

\begin{align*} d(\highlight{9})\amp=5-\frac{5}{4}\abs{\highlight{9}-4}\\ \amp=5-\frac{5}{4}\abs{5}\\ \amp=5-\frac{5}{4}\cdot 5\\ \amp=\frac{20}{4}-\frac{25}{4}\\ \amp=-\frac{5}{4}\\ \amp=-1.25 \end{align*}

So, it looks like if you stand \(9\) feet from the left wall, you need to stand \(1.25\) feet behind the center line (which would be \(6.25\) feet from the wall with the mirror on it) to be hit by the laser.

Absolute value functions are also used when a value must be within a certain distance or tolerance. For example, a person's body temperature is considered “normal” if it is within \(0.5\) degrees of 98.6 °F, so their temperature could be up to \(0.5\) degrees less than or greater than that temperature. To be within normal range, the difference between the two values must be less than or equal to \(0.5\text{,}\) and it does not matter whether it is positive or negative. We will introduce a function for measuring this in the next example.

Example 12.1.12.

The function \(D\) defined by \(D(T)=\abs{T-98.6}\) represents the difference between a person's temperature, T, in Fahrenheit, and 98.6 °F. A person's temperature is considered “normal” if \(D(T)\) is less than or equal to \(0.5\text{.}\) Use \(D(T)\) to determine whether each person's temperature is within the normal range.

LaShonda has a temperature of 98.3 °F.
Castel has a temperature of 99.3 °F.
Daniel has a temperature of 97.3 °F.

Explanation

LaShonda has a temperature of 98.3 °F, so we have:

\begin{align*} D(\highlight{98.3})\amp=\abs{\highlight{98.3}-98.6}\\ \amp=\abs{-0.3}\\ \amp=0.3 \end{align*}

Since the value of \(D(98.3)\) is a number smaller than \(0.5\text{,}\) her temperature of 98.3 °F is within the normal range.
If Castel has a temperature of 99.3 °F, then we have:

\begin{align*} D(\highlight{99.3})\amp=\abs{\highlight{99.3}-98.6}\\ \amp=\abs{0.7}\\ \amp=0.7 \end{align*}

Since the value of \(D(99.3)\) is a number bigger than \(0.5\text{,}\) their temperature of 99.3 °F is not within the normal range.
Daniel's temperature is 97.3 °F, so we have:

\begin{align*} D(\highlight{97.3})\amp=\abs{\highlight{97.3}-98.6}\\ \amp=\abs{-1.3}\\ \amp=1.3 \end{align*}

Since the value of \(D(97.3)\) is a number bigger than \(0.5\text{,}\) his temperature of 97.3 °F is not within the normal range.

Example 12.1.13.

The entryway to the Louvre Museum in Paris is through I. M. Pei's metal and glass Louvre Pyramid. This pyramid has a square base and is \(71\) feet high and \(112\) feet wide. The formula \(h(x)=71-\frac{71}{56}\abs{x-56}\) gives the height above ground level of the pyramid at a distance of \(x\) from the left side of the pyramid base.

A Cartesian graph of y=71-71/56*abs(x-56), which looks like an upside down V from (0,0) to (56, 71) and back down to (112,0). — Figure 12.1.14. A Diagram of the Front of the Louvre Pyramid

If you are \(20\) feet from the left edge, how high will the pyramid rise in front of you? Round your result to the nearest tenth of an inch.
How far from the left edge is the center of the pyramid?
Using your previous answer, check that the formula gives you the correct height at the center.

Explanation

If you are \(20\) feet from the left edge, then \(x\) is \(20\text{.}\) Substituting \(20\) for \(x\) we have

\begin{align*} h(\substitute{20})\amp=71-\frac{71}{56}\abs{\substitute{20}-56}\\ \amp=71-\frac{71}{56}\abs{-36}\\ \amp=71-\frac{71}{56}\cdot 36\\ \amp\approx 71-45.643\\ \amp\approx 25.357\\ \amp\approx 25.4 \end{align*}

The pyramid is about \(25.4\) feet high at the position \(20\) feet from the left edge.
The center of the pyramid is \(56\) feet from the either edge since it's half of \(112\) feet.
Putting \(x=56\) into the formula for \(h\) gives us

\begin{align*} h(\substitute{56})\amp=71-\frac{71}{56}\abs{\substitute{56}-56}\\ \amp=71-\frac{71}{56}\abs{0}\\ \amp=71-\frac{71}{56}\cdot 0\\ \amp=71 \end{align*}

And so the formula does give us the correct maximum height of \(71\) feet at the center of the pyramid.

Exercises 12.1.5 Exercises

Review and Warmup

1.

Evaluate the following.

\(\displaystyle{ \left\lvert 10 \right\rvert = }\)
\(\displaystyle{ \left\lvert -6 \right\rvert = }\)
\(\displaystyle{ \left\lvert 0 \right\rvert = }\)
\(\displaystyle{ \left\lvert {10+\left(-1\right)} \right\rvert = }\)
\(\displaystyle{ \left\lvert {-7-\left(-1\right)} \right\rvert = }\)

2.

Evaluate the following.

\(\displaystyle{ \left\lvert 1 \right\rvert = }\)
\(\displaystyle{ \left\lvert -9 \right\rvert = }\)
\(\displaystyle{ \left\lvert 0 \right\rvert = }\)
\(\displaystyle{ \left\lvert {14+\left(-4\right)} \right\rvert = }\)
\(\displaystyle{ \left\lvert {-8-\left(-3\right)} \right\rvert = }\)

3.

Evaluate the following.

\(\displaystyle{ - \lvert 4-10 \rvert = }\)
\(\displaystyle{ \lvert -4-10 \rvert = }\)
\(\displaystyle{ -4 \lvert 10-4 \rvert = }\)

4.

Evaluate the following.

\(\displaystyle{ - \lvert 3-7 \rvert = }\)
\(\displaystyle{ \lvert -3-7 \rvert = }\)
\(\displaystyle{ -4 \lvert 7-3 \rvert = }\)

5.

Evaluate the following.

\(\displaystyle{ -7-2\left\lvert 8-4\right\rvert = }\)
\(\displaystyle{ -7-2\left\lvert 4-8\right\rvert = }\)

6.

Evaluate the following.

\(\displaystyle{ -6-7\left\lvert 5-3\right\rvert = }\)
\(\displaystyle{ -6-7\left\lvert 3-5\right\rvert = }\)

7.

Evaluate the following.

\(\displaystyle{ -5-4\left\lvert 7-1\right\rvert = }\)
\(\displaystyle{ -5-4\left\lvert 1-7\right\rvert = }\)

8.

Evaluate the following.

\(\displaystyle{ -4-2\left\lvert 9-4\right\rvert = }\)
\(\displaystyle{ -4-2\left\lvert 4-9\right\rvert = }\)

9.

Evaluate the following.

\(\displaystyle{ 5-8\left\lvert 2-7 \right\rvert + 4 = }\)

10.

Evaluate the following.

\(\displaystyle{ 5-7\left\lvert 4-7 \right\rvert + 4 = }\)

11.

Evaluate the following.

\(\displaystyle{ 1-8\left\lvert -9+(2-6)^{3}\right\rvert = }\)

12.

Evaluate the following.

\(\displaystyle{ 2-6\left\lvert -3+(4-6)^{3}\right\rvert = }\)

Function Notation with Absolute Value

13.

Given \(f(r) = \left|r - 280\right|\text{,}\) find and simplify \(f(125)\text{.}\)

\(f(125)={}\)

14.

Given \(F(y) = \left|y - 346\right|\text{,}\) find and simplify \(F(136)\text{.}\)

\(F(136)={}\)

15.

Given \(f(y) = {\left|3y+7\right|}\text{,}\) find and simplify \(f(15)\text{.}\)

\(f(15)={}\)

16.

Given \(H(y) = {\left|-2y-15\right|}\text{,}\) find and simplify \(H(16)\text{.}\)

\(H(16)={}\)

17.

Given \(h(t) = {15-\left|2t+25\right|}\text{,}\) find and simplify \(h(17)\text{.}\)

\(h(17)={}\)

18.

Given \(g(y) = {12-\left|-3y+4\right|}\text{,}\) find and simplify \(g(18)\text{.}\)

\(g(18)={}\)

19.

Given \(g(t) = {t+\left|4t-17\right|}\text{,}\) find and simplify \(g(20)\text{.}\)

\(g(20)={}\)

20.

Given \(G(y) = {y+\left|4y+22\right|}\text{,}\) find and simplify \(G(10)\text{.}\)

\(G(10)={}\)

21.

Given \(h(x) = {\left|x^{2}+x-12\right|}\text{,}\) find and simplify \(h(-2)\text{.}\)

\(h(-2)={}\)

22.

Given \(f(x) = {\left|x^{2}+3x-18\right|}\text{,}\) find and simplify \(f(4)\text{.}\)

\(f(4)={}\)

23.

Given \(H(x) = {\left|x^{2}-x-12\right|}\text{,}\) find and simplify \(H(-5)\text{.}\)

\(H(-5)={}\)

24.

Given \(F(x) = {\left|x^{2}+x-20\right|}\text{,}\) find and simplify \(F(2)\text{.}\)

\(F(2)={}\)

Domain

25.

Find the domain of \(G\) where \(\displaystyle{G(x)=\lvert {-8x-10} \rvert}\text{.}\)

26.

Find the domain of \(G\) where \(\displaystyle{G(x)=\lvert {6x-1} \rvert}\text{.}\)

27.

Find the domain of \(H\) where \(\displaystyle{H(x) = 8x - \lvert {-x+8} \rvert}\text{.}\)

28.

Find the domain of \(K\) where \(\displaystyle{K(x) = 5x - \lvert {-9x-5} \rvert}\text{.}\)

Tables

29.

Make a table of values for the function \(f\) defined by \(f(x)={\left|3x-3\right|}\text{.}\)

\(x\)	\(f(x)\)

30.

Make a table of values for the function \(f\) defined by \(f(x)={\left|3x-3\right|}\text{.}\)

\(x\)	\(f(x)\)

31.

Make a table of values for the function \(g\) defined by \(g(x)={\left|x^{2}-2x-3\right|}\text{.}\)

\(x\)	\(g(x)\)

32.

Make a table of values for the function \(h\) defined by \(h(x)={\left|x^{2}-x-3\right|}\text{.}\)

\(x\)	\(h(x)\)

33.

Make a table of values for the function \(F\) defined by \(F(x)={2\!\left|3x-2\right|+3}\text{.}\)

\(x\)	\(F(x)\)

34.

Make a table of values for the function \(G\) defined by \(G(x)={2\!\left|-3x+2\right|-3}\text{.}\)

\(x\)	\(G(x)\)

35.

Make a table of values for the function \(G\) defined by \(G(x)=\Big\lvert{-3\!\left|x-2\right|+3}\Big\rvert\text{.}\)

\(x\)	\(G(x)\)

36.

Make a table of values for the function \(H\) defined by \(H(x)=\Big\lvert{-2\!\left|x-2\right|+3}\Big\rvert\text{.}\)

\(x\)	\(H(x)\)

Graphs

37.

Graph \(y=f(x)\text{,}\) where \(f(x)=\left\lvert2x-1\right\rvert\text{.}\)

38.

Graph \(y=f(x)\text{,}\) where \(f(x)=\left\lvert x-2\right\rvert\text{.}\)

39.

Graph \(y=f(x)\text{,}\) where \(f(x)=\left\lvert x^2-2x-1\right\rvert\text{.}\)

40.

Graph \(y=f(x)\text{,}\) where \(f(x)=\left\lvert x^2+3x-2\right\rvert\text{.}\)

41.

Graph \(y=f(x)\text{,}\) where \(f(x)=\frac{1}{2}\left\lvert 4x-5\right\rvert-3\text{.}\)

42.

Graph \(y=f(x)\text{,}\) where \(f(x)=\frac{3}{4}\left\lvert 6+x\right\rvert+2\text{.}\)

43.

Graph \(y=f(x)\text{,}\) where \(f(x)=\big\lvert2\left\lvert3-x\right\rvert-2\big\rvert\text{.}\)

44.

Graph \(y=f(x)\text{,}\) where \(f(x)=\big\lvert3 - 2\left\lvert2x - 3\right\rvert\big\rvert\text{.}\)

Applications

45.

The height inside a camping tent when you are \(d\) feet from the edge of the tent is given by

\begin{equation*} h={-1.1\!\left|d-4\right|+3.5} \end{equation*}

where \(h\) stands for height in feet.

Determine the height when you are:

\({6.3\ {\rm ft}}\) from the edge.

The height inside a camping tent when you are \({6.3\ {\rm ft}}\) from the edge of the tent is .
\({2.4\ {\rm ft}}\) from the edge.

The height inside a camping tent when you are \({2.4\ {\rm ft}}\) from the edge of the tent is .

46.

The height inside a camping tent when you are \(d\) feet from the edge of the tent is given by

\begin{equation*} h={-0.9\!\left|d-5.2\right|+3.5} \end{equation*}

where \(h\) stands for height in feet.

Determine the height when you are:

\({8\ {\rm ft}}\) from the edge.

The height inside a camping tent when you are \({8\ {\rm ft}}\) from the edge of the tent is .
\({2.7\ {\rm ft}}\) from the edge.

The height inside a camping tent when you are \({2.7\ {\rm ft}}\) from the edge of the tent is .

47.

Write two numbers so that

The first number is less than the second number, and
The absolute value of the first number is greater than the absolute value of the second number

and