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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.

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

Solution

To identify the vertex of the graph, the given function needs to be written in vertex form.

$2y+1=\text{-}4|x-1|+9$

Write in vertex form

SubEqn

$\text{LHS}-1=\text{RHS}-1$

$2y=\text{-}4|x-1|+8$

FactorOut

Factor out $2$

$2y=2(\text{-}2|x-1|+4)$

DivEqn

$\text{LHS}/2=\text{RHS}/2$

$y=\text{-}2|x-1|+4$

Now that the function is in vertex form, the vertex of the graph can be identified.

$$\begin{array}{cc}\underline{\text{Vertex Form}}& \underline{\text{Vertex}}\\ y=\text{-}2|x-1|+4& (1,4)\end{array}$$

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.

$x$	$y=\text{-}2|x-1|+4$	$y$
$\text{-}5$	$y=\text{-}2|\text{-}5-1|+4$	$\text{-}8$
$\text{-}3$	$y=\text{-}2|\text{-}3-1|+4$	$\text{-}4$
$\text{-}1$	$y=\text{-}2|\text{-}1-1|+4$	$0$
$1$	$y=\text{-}2|1-1|+4$	$4$
$3$	$y=\text{-}2|3-1|+4$	$0$
$5$	$y=\text{-}2|5-1|+4$	$\text{-}4$
$7$	$y=\text{-}2|7-1|+4$	$\text{-}8$

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 $x\text{-}$intercepts at $(\text{-}1,0)$ and $(3,0),$ respectively, and one $y\text{-}$intercept at $(0,2).$

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

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.

Solution

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.

$$\begin{array}{c}h=71-\frac{71}{56}|x-56|\\ \Updownarrow \\ h=\text{-}\frac{71}{56}|x-56|+71\end{array}$$

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.

$x$	$h=71-\frac{71}{56}|x-56|$	$h$
$0$	$h=71-\frac{71}{56}|0-56|$	$0$
$20$	$h=71-\frac{71}{56}|20-56|$	$\approx 25.36$
$40$	$h=71-\frac{71}{56}|40-56|$	$\approx 50.71$
$56$	$h=71-\frac{71}{56}|56-56|$	$71$
$70$	$h=71-\frac{71}{56}|60-56|$	$\approx 53.25$
$90$	$h=71-\frac{71}{56}|80-56|$	$\approx 27.89$
$110$	$h=71-\frac{71}{56}|100-56|$	$\approx 2.54$

Plot the ordered pairs from the table and graph the function.

Looking at the graph, the $x\text{-}$intercepts are the points where the pyramid's height is $0$ feet. This means that the $x\text{-}$intercepts are on the opposite sides of the pyramid. Since the pyramid has a square base, the distance between the $x\text{-}$intercepts will be the width of the pyramid.

Looking at the table and the graph, it can be seen that the $x\text{-}$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 $x\text{-}$intercept should be at $(112,0).$ This means that the width of the pyramid is $112$ feet.

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.

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.

Solution

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.

$$\begin{array}{c}y=12|x-5|-3\\ \Updownarrow \\ y=12|x-5|+(\text{-}3)\end{array}$$

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

$$\begin{array}{c}12>0\end{array}$$

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.

$x$	$y=12|x-5|-3$	$y$
$0$	$y=12|0-5|-3$	$57$
$2$	$y=12|2-5|-3$	$33$
$4$	$y=12|4-5|-3$	$9$
$5$	$y=12|5-5|-3$	$\text{-}3$
$6$	$y=12|6-5|-3$	$9$
$8$	$y=12|8-5|-3$	$33$
$10$	$y=12|10-5|-3$	$57$

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 $y\text{-}$intercept shows the height where the brown pelican starts its dive. From the table, it can be seen that the $y\text{-}$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 $x\text{-}$axis represents the surface of the sea. The $x\text{-}$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.

By finding the $x\text{-}$intercepts of the graph, the time spent under the water can be found. First substitute $y=0$ into the function and isolate the absolute value expression.

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

Substitute

$y=0$

$0=12|x-5|-3$

Isolate absolute value expression

AddEqn

$\text{LHS}+3=\text{RHS}+3$

$3=12|x-5|$

DivEqn

$\text{LHS}/12=\text{RHS}/12$

$\frac{3}{12}=|x-5|$

ReduceFrac

$\frac{a}{b}=\frac{a/3}{b/3}$

$\frac{1}{4}=|x-5|$

WriteDec

Write as a decimal

$0.25=|x-5|$

RearrangeEqn

Rearrange equation

$|x-5|=0.25$

Now that the absolute value expression has been isolated, it can be solved for $x.$

$|x-5|=0.25$

$\begin{array}{lc}x-5\ge 0\text{: }x-5=0.25& \text{(I)}\\ x-5<0\text{: }x-5=\text{-}0.25& \text{(II)}\end{array}$

$\begin{array}{lc}x-5=0.25& \text{(I)}\\ x-5=\text{-}0.25& \text{(II)}\end{array}$

$\text{(I), (II): }$$\text{LHS}+5=\text{RHS}+5$

$\begin{array}{l}x=5.25\\ x=4.75\end{array}$

The $x\text{-}$intercepts of the function are $(5.25,0)$ and $(4.75,0).$ With this information, the time difference between under water (in) and above water (out) can be solved.

$$\begin{array}{c}5.25-4.75=0.5\text{ s}\end{array}$$

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