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This lesson will focus on graphing absolute value functions through the use of a table of values. Afterward, the key features of absolute value graphs will be examined.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

*Absolute value functions* can be used to model several situations. Consider the amazing movements of diving birds such as the Brown Pelican that dives headfirst into the ocean surf to snatch small fish!

The following absolute value function represents the distance traveled by the brown pelican at any time during its dive, from start to finish.

$y=12∣x−5∣−3 $

In this equation, $y$ is the distance in feet and $x$ is the time in seconds. The brown pelican starts its descent at $x=0.$

a How deep beyond the surface of the ocean does the brown pelican dive?

b From what height does the brown pelican start its dive to catch fish?

c How many seconds does the brown pelican spend under water?

Discussion

In order to draw the graphs of absolute value functions and interpret them, the concept of absolute value function needs to be understood. Well, what exactly is an absolute value function?

Concept

An absolute value function is a function that contains an absolute value expression.

$y=∣x−1∣+1 $

An absolute value graph has a distinct $V-$shape. It is symmetric about the vertical line that passes through its vertex. The vertex of an absolute value function is the point where the graph changes direction.

Since the graph of $y=∣x−1∣+1$ changes direction at $(1,1),$ this point is the vertex. Additionally, the graph is symmetric about the line $x=1.$The graph of an absolute value function can be drawn using a table of values.

Method

Making a table of values is the most basic method to graph an absolute value function. The following equation will be drawn as an example.
*expand_more*
*expand_more*

*expand_more*

$2y=4∣x−2∣+6 $

To draw the graph of the equation, the following steps can be followed.
1

Isolate the $y-$Variable and Simplify the Equation

2

Make a Table of Values

Now that the equation has been simplified, a table of values can be made.

$x$ | $y=2∣x−2∣+3$ | $y$ |
---|---|---|

$-4$ | $y=2∣-4−2∣+3$ | $15$ |

$-2$ | $y=2∣-2−2∣+3$ | $11$ |

$0$ | $y=2∣0−2∣+3$ | $7$ |

$2$ | $y=2∣2−2∣+3$ | $3$ |

$4$ | $y=2∣4−2∣+3$ | $7$ |

$6$ | $y=2∣6−2∣+3$ | $11$ |

$8$ | $y=2∣8−2∣+3$ | $15$ |

3

Plot the Ordered Points and Graph the Function

Finally, plot the ordered pairs from the table of values and connect them.

Example

Paulina is excited to study the sales of a shoe called Absolutes that she wants to buy and wear for when she plays pool. Over a certain time, the sales of Absolutes increased, followed by a sharp decrease. The following absolute value function expresses the shoe's market performance.

$s=-2∣t−10∣+40 $

In this equation, $s$ is the number of shoes sold in the thousands and $t$ is the time in weeks. Draw the graph of this sale by making a table of values so Zain can show their investors. Is the given absolute value function in its simplest form?

Notice that the given absolute value function is already in its simplest form.

$s=-2∣t−10∣+40 $

Therefore, a table of values can be made already. Since time cannot be negative, $t$ will be greater than or equal to $0.$

$t$ | $s=-2∣t−10∣+40$ | $s$ |
---|---|---|

$0$ | $s=-2∣0−10∣+40$ | $20$ |

$5$ | $s=-2∣5−10∣+40$ | $30$ |

$10$ | $s=-2∣10−10∣+40$ | $40$ |

$15$ | $s=-2∣15−10∣+40$ | $30$ |

$20$ | $s=-2∣20−10∣+40$ | $20$ |

$25$ | $s=-2∣25−10∣+40$ | $10$ |

$30$ | $s=-2∣30−10∣+40$ | $0$ |

Now that the ordered pairs have been determined, they can be plotted on the coordinate plane and the graph of the sales can be drawn.

Paulina ia stunned that the sales of her favorite shoe have gone down. Maybe she will be able to get them at a discounted price!

Example

Paulina, wearing the new Absolutes for good luck, is ready to play pool against her archrival Vincenzo. Thanks to a better understanding of absolute value functions, however, she does not need luck. She can apply some of her knowledge to calculate the game-winning shot.

Her plan is to bank the five-ball off the side represented by the $x-$axis. The path of the ball is described by the following absolute value function.

$y=23 ∣x−2∣ $

With the given path, is Paulina able to make the shot? Explain.

Yes, see solution.

Draw the graph of the equation by making a table of values and then interpret the graph.

To decide whether Paulina makes the shot using the given path, the graph of the function needs be drawn. To do so, make a table of values.

$x$ | $y=23 ∣x−2∣$ | $y$ |
---|---|---|

$-2$ | $y=23 ∣-2−2∣$ | $6$ |

$0$ | $y=23 ∣0−2∣$ | $3$ |

$2$ | $y=23 ∣2−2∣$ | $0$ |

$4$ | $y=23 ∣4−2∣$ | $3$ |

$6$ | $y=23 ∣6−2∣$ | $6$ |

Next, plot the ordered pairs on the given coordinate plane and draw the graph.

The five ball goes into the pocket at $(6,6),$ and Paulina makes the shot to beat her archrival!

Discussion

Now that graphing an absolute value function using a table of values has been covered, from now on, characteristics of absolute value graphs will be interpreted. The most characteristic feature of an absolute value graph is its vertex. By writing the absolute value function in *vertex form*, its vertex can be identified immediately.

Concept

The vertex form of an absolute value function is written as follows.

$y=a∣x−h∣+k $

In this form, $a =0$ and $(h,k)$ is the vertex of the graph of the function. Any absolute value function can be written in vertex form, and its graph is symmetric about the line $x=h.$ Based on the value of $a,$ the vertex of an absolute value graph can be either the *maximum value* or the *minimum value* of the graph.

- If the value of $a$ is greater than $0,$ then the absolute value graph opens upward and the vertex becomes the
**minimum**value.

- If the value of $a$ is less than $0,$ then the absolute value graph opens downward and the vertex becomes the
**maximum**value.

See the following graphs for reference.

Along with the value of $a,$ the value of $k$ affects the number of $x-$intercepts of an absolute value graph. Note that an absolute value graph always has one $y-$intercept. However, it can have zero, one, or two $x-$intercepts.

Pop Quiz

Considering the characteristics of an absolute value function graph, identify the $x-$intercepts, $y-$intercept, maximum, and minimum of the given graphs. Round the answers to two decimal places if it is needed.

Example

After seeing how great algebra works when playing billiards, Paulina wants to better understand another absolute value function. She might be able to apply this to future games.

$2y+1=-4∣x−1∣+9 $

She wants to identify the key features of the graph of this function. Help her find the vertex, $y-$intercept, and $x-$intercepts of the graph.

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Begin by identifying the vertex of the function and then graph it using a table of values.

To identify the vertex of the graph, the given function needs to be written in vertex form.
Now that the function is in vertex form, the vertex of the graph can be identified.

$2y+1=-4∣x−1∣+9$

$y=-2∣x−1∣+4$

$Vertex Form y=-2∣x−1∣+4 Vertex (1,4) $

From here, by making a table of values, the graph of the function can be drawn. Note that the graph will be symmetric about the line $x=1.$ Therefore, the vertex will be helpful to identify the values in the table symmetrically.

$x$ | $y=-2∣x−1∣+4$ | $y$ |
---|---|---|

$-5$ | $y=-2∣-5−1∣+4$ | $-8$ |

$-3$ | $y=-2∣-3−1∣+4$ | $-4$ |

$-1$ | $y=-2∣-1−1∣+4$ | $0$ |

$1$ | $y=-2∣1−1∣+4$ | $4$ |

$3$ | $y=-2∣3−1∣+4$ | $0$ |

$5$ | $y=-2∣5−1∣+4$ | $-4$ |

$7$ | $y=-2∣7−1∣+4$ | $-8$ |

Now that the ordered pairs are identified, plot them on the coordinate plane and draw the graph.

As it can be seen, the graph has two $x-$intercepts at $(-1,0)$ and $(3,0),$ respectively, and one $y-$intercept at $(0,2).$

With this knowledge, Paulina is on her way to improving her game for a long time to come.

Example

Vincenzo is feeling sad about losing his billiards match to an archrival. In math class, thinking of ways to improve, Vincenzo finds himself staring at this picture of the Louvre Museum in Paris.

Suddenly, he remembers how his archrival used algebra to improve their skills. He can learn algebra also! He will start now. Feeling energized, he sees that the book gives the absolute value function where $h$ is the height in feet above ground level of the pyramid at a distance of $x$ feet from the left side of the pyramid's base. The pyramid has a square base.

$h=71−5671 ∣x−56∣ $

a What is the height of the pyramid?

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b What is the width of the pyramid?

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b What do the $x-$intercepts of the function represent in this context?

a Recall that the vertex of an absolute value function represents either the maximum or minimum value of the function. Write the function in vertex form.

$h=71−5671 ∣x−56∣⇕h=-5671 ∣x−56∣+71 $

The vertex of the function is $(56,71).$ Since the value of $a$ is less than $0,$ the function has a maximum at its vertex. The height of the pyramid is $71$ feet. b To find the width of the pyramid, the given function can be used. Graph the function using a table of values. Since the distance $x$ is a *linear measurement*, it cannot be negative.

Plot the ordered pairs from the table and graph the function.
Looking at the graph, the $x-$intercepts are the points where the pyramid's height is $0$ feet. This means that the $x-$intercepts are on the opposite sides of the pyramid. Since the pyramid has a square base, the distance between the $x-$intercepts will be the width of the pyramid.

$x$ | $h=71−5671 ∣x−56∣$ | $h$ |
---|---|---|

$0$ | $h=71−5671 ∣0−56∣$ | $0$ |

$20$ | $h=71−5671 ∣20−56∣$ | $≈25.36$ |

$40$ | $h=71−5671 ∣40−56∣$ | $≈50.71$ |

$56$ | $h=71−5671 ∣56−56∣$ | $71$ |

$70$ | $h=71−5671 ∣60−56∣$ | $≈53.25$ |

$90$ | $h=71−5671 ∣80−56∣$ | $≈27.89$ |

$110$ | $h=71−5671 ∣100−56∣$ | $≈2.54$ |

Looking at the table and the graph, it can be seen that the $x-$intercept on the left side is $(0,0).$ Recall that the graph of an absolute value function is symmetric about the vertical line that passes through its vertex. In this case it is symmetric about the line $x=56.$

The other $x-$intercept should be at $(112,0).$ This means that the width of the pyramid is $112$ feet.Closure

Considering the concepts and methods covered throughout the lesson, the challenge provided at the beginning can be solved. Presented was the unique movement of a brown pelican, known to be a diving bird, that is able to dive head first into the ocean surf to catch small fish. An absolute value function was used to model this amazing feat.

External credits: Charles J. Sharp

The following absolute value function shows the distance of a brown pelican from the surface of the sea.

$y=12∣x−5∣−3 $

In this equation, $y$ is the distance in feet and $x$ is the time in seconds. The brown pelican starts its descent at $x=0$ to catch fish.

a How deep beyond the surface of the ocean does the brown pelican dive to catch fish?

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b From what height does the brown pelican start its dive to catch fish?

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c How many seconds does the brown pelican spend under water?

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b Identify and interpret the $y-$intercept of the function.

c Identify and interpret the $x-$intercepts of the function.

a Recall that the vertex of an absolute value function represents the maximum or the minimum value of the function. By identifying the vertex of the function, the depth of the dive can be determined. First, rewrite the given function in vertex form.

$y=12∣x−5∣−3⇕y=12∣x−5∣+(-3) $

In this form, the vertex can be identified as $(5,-3).$ In addition to this, notice that the coefficient of the absolute value expression is positive.
$12>0 $

Therefore, the given function has a minimum at the vertex. In this context, the vertex of the function can be interpreted as the brown pelican diving $3$ feet deep after diving for $5$ seconds.
b To find the height in air that the dive starts from, first draw the graph of the function by making a table of values. Note that since $x$ represents the time, it cannot be negative.

$x$ | $y=12∣x−5∣−3$ | $y$ |
---|---|---|

$0$ | $y=12∣0−5∣−3$ | $57$ |

$2$ | $y=12∣2−5∣−3$ | $33$ |

$4$ | $y=12∣4−5∣−3$ | $9$ |

$5$ | $y=12∣5−5∣−3$ | $-3$ |

$6$ | $y=12∣6−5∣−3$ | $9$ |

$8$ | $y=12∣8−5∣−3$ | $33$ |

$10$ | $y=12∣10−5∣−3$ | $57$ |

From here, plot the ordered pairs on the coordinate plane and draw the graph.

The graph shows that the height of the brown pelican starts to decrease after $x=0.$ This means that the $y-$intercept shows the height where the brown pelican starts its dive. From the table, it can be seen that the $y-$intercept is $(0,57).$ Therefore, its starting height is $57$ feet above the surface of the ocean.

c Interpreting the graph drawn in Part B, it can be concluded that the $x-$axis represents the surface of the sea. The $x-$intercept on the left shows the time that the brown pelican enters the water. The one on the right shows the time that the brown pelican comes out of the water.

$y=12∣x−5∣−3$

Substitute

$y=0$

$0=12∣x−5∣−3$

Isolate absolute value expression

AddEqn

$LHS+3=RHS+3$

$3=12∣x−5∣$

DivEqn

$LHS/12=RHS/12$

$123 =∣x−5∣$

ReduceFrac

$ba =b/3a/3 $

$41 =∣x−5∣$

WriteDec

Write as a decimal

$0.25=∣x−5∣$

RearrangeEqn

Rearrange equation

$∣x−5∣=0.25$

$∣x−5∣=0.25$

$x−5≥0:x−5=0.25x−5<0:x−5=-0.25 (I)(II) $

$x−5=0.25x−5=-0.25 (I)(II) $

$(I), (II):$$LHS+5=RHS+5$

$x=5.25x=4.75 $

$5.25−4.75=0.5s $

The results are in! The brown pelican dives underwater for merely $0.5$ seconds. Now that is a quick catch.