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Every day people have problems that the application of linear functions can solve. In this lesson, some common situations will be presented and solved seamlessly using them.
### Catch-Up and Review

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

Challenge

Dominika and her friend Zosia live relatively close to each other. Each morning they leave their homes to go to school, and at some point, they meet and continue the trip together. Usually, when walking separately, they walk at different speeds.
The following two linear functions describe the way the girls walk to school separately.

External credits: @pikisuperstar

$f(x)=5x+aandg(x)=3x+b $

Here, $x$ denotes the time the girls are walking, while $f$ and $g$ represent each girls' distance from Dominika's home. The graphs of these functions and their point of intersection are demonstrated in the applet. a If Dominika walks faster than Zosia, which function corresponds to her? Label the graphs of $f$ and $g$ on the coordinate plane.

b Locate $a$ and $b$ on the coordinate plane. Then, interpret those values in terms of the given situation.

c Consider the points with the $x-$coordinates of $m+2$ on the functions $f$ and $g,$ where $(m,n)$ is their point of intersection. What is the difference between the $y-$coordinates of these points?

Example

Dominika, a budding citizen scientist, is curious about the changes in temperature where she lives in Phoenix, Arizona. Since she walks to school everyday, the temperature affects her walk. Dominika decides to measure the temperature at three different times on the same day. Here are the measurements she recorded.

Time | Temperature |
---|---|

$16:00$ | $94_{∘}$F |

$19:00$ | $88_{∘}$F |

$21:00$ | $78_{∘}$F |

When was the temperature decreasing the fastest: between $16:00$ and $19:00$ or between $19:00$ and $21:00?$

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Calculate the change in time and temperature. Use the formula for the rate of change.

In order to determine when the temperature was decreasing the fastest, the rate of change should be calculated. First, recall its definition.

$Rate of change=ΔxΔy $

In this formula, $Δy$ represents the change in the output variable and $Δx$ the change in the input variable. Because the change of temperature $T$ over the change of time $t$ needs to be found, the formula for the rate of change can be rewritten in terms of $T$ and $t.$
$Rate of change=ΔtΔT $

To apply this formula, the changes in time and temperature will be calculated. Time Interval | $Δt$ | $ΔT$ |
---|---|---|

$16:00−19:00$ | $3$ hours | $-6_{∘}F∣∣∣∣ 94−88=$ |

$19:00−21:00$ | $2$ hours | $-10_{∘}F∣∣∣∣ 88−78=$ |

Now, substitute the obtained values of $ΔT$ and $Δt$ into the formula to find the rate of change of each time interval.

Time Interval | $Δt$ | $ΔT$ | Rate of Change |
---|---|---|---|

$16:00−19:00$ | $3$ hours | $-6_{∘}F$ | $3-6 =-2_{∘}F$ per hour |

$19:00−21:00$ | $2$ hours | $-10_{∘}F$ | $2-10 =-5_{∘}F$ per hour |

As can be seen, the temperature decreased at an average rate of $2_{∘}F$ in the first time interval and $5_{∘}F$ in the second time interval. Therefore, the temperature decreased the fastest between $19:00$ and $21:00.$

Example

Along the walk to school, Dominika passes her father's $24-hour$ cafe. Running the cafe requires close attention to customer behavior. Her father notices that it is typical to serve $1$ customer each night from $2:00AM$ till $4:00AM.$ Then, starting from $6:00AM,$ the cafe slowly begins to fill with regulars.

Let $P$ represent the number of people in the cafe and $t$ represent the time in hours since midnight.

a Is $t$ a function of $P?$ Explain your answer.

b Is $P$ a function of $t?$ If yes, consider a function $P=f(t)$ and interpret the meaning of $f(3)$ in the context of the given situation.

c Mr. Beckett noticed that the number of customers usually increases by an average of $20$ people during the lunch hours from $12:00PM$ to $2:00PM.$ Which of the following equations best represents that observation?

$I.II.III.IV. f(12)=20f(14)−f(12)=202f(14)−f(12) =20P=f(12)+20 $

a No, because the input value of $P=1$ is associated with three output values $t=1,$ $t=2,$ $t=3.$

b Yes, $f(3)$ represents the number of customers in the cafe at $3:00AM.$

c $(ii)f(14)−f(12)=20$

a Recall the definition of a function. Investigate whether each input is assigned to exactly one output.

b To interpret the meaning of $f(3),$ think about what variables $t$ and $P$ denote.

c Analyze the meaning of each given equation. How can the number of customers at $12:00PM$ and $2:00PM$ be written using a function notation?

a To determine whether $t$ is a function of $P,$ recall the definition of a function.

In the diagram, it can be observed that there would be more than one output $t$ associated with the single input value of $P=1.$ This fact contradicts the definition of a function. Therefore, $t$ cannot be a function of $P.$

$t=f(P)× $

b Contrary to Part A, if $P$ is a function of $t,$ each input $t$ is assigned to exactly one output $P.$

Note that it does not matter that different input values are assigned to the same output value. The important thing is that they are not assigned to

$P=f(t)✓ $

To interpret the meaning of $f(3),$ recall that the input $t$ represents the time in hours since midnight, while the output $P$ represents the number of people in the cafe.
The expression $f(3)$ represents the corresponding value of $P$ when $t=3.$ In this situation, $t=3$ represents $3:00AM$ and $f(3)$ is the number of people in the cafe at that time.

c In order to identify the equation which best describes Mr. Beckett's observation, each given equation will be analyzed separately.

Equation | Meaning |
---|---|

$f(12)=20$ | At $12:00PM,$ there are $20$ customers in the cafe. |

$f(14)−f(12)=20$ | The difference between the number of customers at $12:00PM$ and the number of customers at $2:00PM$ is $20$ people. |

$2f(14)−f(12) =20$ | The average change of the number of customers per hour between $12:00PM$ and $2:00PM$ is $20$ people. |

$P=f(12)+20$ | The number of people at the cafe equals the number of customers at $12:00PM$ plus $20$ more customers. |

The second equation can be rephrased as the number of customers at $2:00PM$ increased by $20$ since $12:00PM,$ which coincides with what Mr. Beckett observed. Therefore, this equation best represents the described situation.

Example

While walking to school, Zosia was thinking about a story her auntie just told her. In the windy city crazy winds, flurries of snow, and a max capacity of $61500$ cheering fans have been a part of the legacy of Soldier Field, where the Chicago Bears have played on the gridiron each Sunday for decades. The average cost of ticket to a game was $$45$ at some point in time.

Zosia's auntie went to a game against there rivals, only to hear that the owner was thinking of selling the team. If only they could make more money from ticket sales! The amount of money earned from the game her auntie attended is a function of the number of people $n$ who attended it.

a What is the domain of this function? Write the answer as a compound inequality.

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b What is the range of this function if $M$ is the amount of money earned from the game? Write the answer as a compound inequality.

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c Write the function for the average profit $P$ from the game if the cost of organizing the game is $$100000.$

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a Analyze what minimum and maximum values $n$ can have.

c First, find the expression for the earnings from the game. Then subtract the cost of holding the game.

a It is given that the amount of money earned from the game is a function of the number of people $n$ who will attend the game. Let $M$ be the amount of money earned from the game. Now, the described relation can be written as follows.

$M=f(n) $

Recall that a domain is the set of all inputs for which the function is defined. In this case, the inputs of the function are the values of $n.$ Since $n$ represents the number of people attending the game, it can only have non-negative values.
$n≥0 $

Also, Soldier Field is said to have a maximum capacity of $61500.$ Therefore, $n$ cannot have values greater than $61500.$
$n≤61500 $

Additionally, since $n$ is the number of people, it can only have integer values. Gathering all the found information, the domain of the function can now be determined.
$D={n∈Z:0≤n≤61500} $

b The range of a function is the set of all the outputs of the function. In this case, the outputs consist of all $M$ values, representing the amount of money earned from selling tickets. If nobody attended the game — meaning no tickets were sold — the earnings would be $0.$ This is the minimum value that $M$ could have.

$M≥0 $

In the opposite situation, the stadium would be completely full. This would mean that all tickets were sold, and the earnings would be the maximum possible. In average, a ticket costs $$45,$ and the stadium's total capacity is $61500,$ so the product of these values will give the maximum possible earnings.
$$45⋅61500=$2767500⇓M≤2767500 $

Finally, because both the price of a ticket and the number of people are integers, $M$ will also be an integer number. With this information, the range of the function can now be summarized. $R={M∈Z:0≤M≤2767500} $

c Because the cost of organizing the game is $$100000,$ the profit $P$
can be determined by subtracting this quantity from the money earned by selling the tickets. Note that the money $M$ earned by selling the tickets is equal to the product of the price of a ticket and the number of people $n$ that will attend the game.

$M=45n $

Now, the cost of organizing the game can be subtracted from the money earned to get the profit. $P=45n−100000 $

It can be seen that $P$ is a function of $n.$ This can be explicitly indicated by writing the equation for the average profit in function notation. $P(n)=45n−100000 $

Example

As Dominika was studying temperatures previously, she came across the temperature scales of both Fahrenheit and Celsius. She saw the Kelvin temperature scale — often used in scientific research — for the first time. She finds an online program to make a simple app that converts Kelvins to Fahrenheit and Fahrenheit to Kelvins.
### Hint

### Solution

Now that the value of $b$ is found, substitute it into the above formula of a linear function.
Finally, the function that converts Kelvins into degrees Fahrenheit can be completed.
Therefore, the inverse function of $f$ has been found. In the context of this situation, this inverse function converts degrees Fahrenheit into Kelvins.

At the moment, she only knows two facts that she has already added to the app.

- The zero point on the Kelvin scale is the lowest possible temperature in the universe and is equal to $-459.67_{∘}F.$
- The body temperature in Kelvins is $310.15,$ while in degrees Fahrenheit, it is $98.6.$

Help Dominika find the needed functions so that she can finish her app.

a Write a linear function $f$ that assigns to a temperature in Kelvins its equivalent in degrees Fahrenheit.

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b Find the inverse of the function $f$ and explain its meaning in terms of temperature conversions.

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a Since $f$ is a linear function, it has the form of $y=ax+b.$ Use the given information about the zero point and body temperature to find the values of $a$ and $b.$

b To find the inverse of a function, switch the $x$ and $y$ places, and then solve the equation for $y.$

c If there is a temperature which is the same in Kelvins and degrees Fahrenheit, then $f(x)$ would be equal to $x$ for that value.

a This part asks for a linear function that converts a given temperature in Kelvins into degrees Fahrenheit. The following expression represents the general form of a linear function.

$f(x)=ax+b $

For this situation, the inputs $x$ will represent a temperature in Kelvins and the outputs $f(x)$ will represent the corresponding temperature in degrees Fahrenheit. It is known that $0$ Kelvin equals $-459.67_{∘}F.$ Therefore, the value of the constant $b$ can be calculated by substituting $0$ for $x$ and $-459.67$ for $f(x).$ $f(x)=ax+b$

SubstituteII

$x=0$, $f(x)=-459.67$

$-459.67=a(0)+b$

Solve for $b$

ZeroPropMult

Zero Property of Multiplication

$-459.67=0+b$

IdPropAdd

Identity Property of Addition

$-459.67=b$

RearrangeEqn

Rearrange equation

$b=-459.67$

$f(x)=ax+b⇓f(x)=ax−459.67 $

Using a similar reasoning, the given body temperatures in Kelvin and degrees Fahrenheit can be used to find the value of $a.$ This time, $310.15$ will be substituted for $x$ and $98.6$ for $f(x)$ into the partial function found previously. $f(x)=ax−459.67$

SubstituteII

$x=310.15$, $f(x)=98.6$

$98.6=a(310.15)−459.67$

Solve for $a$

AddEqn

$LHS+459.67=RHS+459.67$

$558.27=310.15a$

RearrangeEqn

Rearrange equation

$310.15a=558.27$

DivEqn

$LHS/310.15=RHS/310.15$

$a=1.8$

$f(x)=1.8x−459.67 $

b In order to find the inverse of the function $f,$ first replace $f(x)$ with $y,$ as $f(x)=y$ describes the input-output relationship of the function.

$y=1.8x−459.67 $

Next, because the inverse of a function reverses $x$ and $y,$ switch $x$ and $y$ places in the function rule.
$x=1.8y−459.67 $

Now, the obtained equation should be solved for $y.$
$x=1.8y−459.67$

Solve for $y$

AddEqn

$LHS+459.67=RHS+459.67$

$x+459.67=1.8y$

Rewrite

Rewrite $1.8$ as $1018 $

$x+459.67=1018 y$

ReduceFrac

$ba =b/2a/2 $

$x+459.67=59 y$

MultEqn

$LHS⋅95 =RHS⋅95 $

$95 (x+459.67)=y$

RearrangeEqn

Rearrange equation

$y=95 (x+459.67)$

$f_{-1}(x)=95 (x+459.67) $

Dominika can now use these two functions to implement the desired feature on her amazing app. $f(x)=x $

Therefore, substitute $f(x)$ with $x$ and solve the equation using inverse operations.
It can be concluded that $574.6$ Kelvins equals $574.6_{∘}F.$
Example

Dominika's father, the chef, knows that his daughter and her friends love to stop by his cafe along their walk home from school on Fridays. He is eager to expand his menu and add flor de calabaza quesadillas because it is his daughter's favorite!

Her father has calculated that there would be a weekly fixed cost of $$75$ for adding a new dish plus an additional $$0.7$ for the ingredients per quesadilla. The total cost $C$ for preparing $x$ quesadillas can be represented by a linear function.$C(x)=0.7x+75 $

Now, the chef needs to figure out how much to charge for one quesadilla. Ideally, he wants to earn $$0.30$ per each quesadilla sold. a If the chef sells $25$ quesadilla per week, what will be the cost of making one quesadilla? At this cost, what should the price of one quesadilla be?

b If the chef wants to be sure that the price of one quesadilla is not greater than $$1.50.$ How many quesadillas should he sell every day to set such a price and still make the desired profit?

c Complete the table showing the cost of making one quesadilla depending on how many quesadillas the chef sells.

Number of Quesadillas Sold | $10$ | $25$ | $50$ | $75$ | $100$ | $200$ |
---|---|---|---|---|---|---|

Total Cost | ||||||

Cost per Quesadilla | ||||||

Sales Price per Quesadilla |

d Explain why the price of a quesadilla becomes smaller as the number of quesadillas sold increases.

a **Cost per Quesadilla:** $$3.70$

**Price of One Quesadilla:** $$4$

b $$1.50$

c

Number of Quesadillas Sold | $10$ | $25$ | $50$ | $75$ | $100$ | $200$ |
---|---|---|---|---|---|---|

Total Cost | $$82$ | $$92.50$ | $$110$ | $$127.50$ | $$145$ | $$215$ |

Cost per Quesadilla | $$8.20$ | $$3.70$ | $$2.20$ | $$1.70$ | $$1.45$ | $$1.08$ |

Sales Price per Quesadilla | $$8.50$ | $$4.00$ | $$2.50$ | $$2.00$ | $$1.75$ | $$1.38$ |

d As more quesadillas are sold, the fixed weekly cost is shared by more quesadillas, which lowers the average cost of making one quesadillas.

a Calculate the total cost of making $25$ quesadillas and then add $$0.30$ to find the price of one quesadilla.

b Find the expression for the average cost per quesadilla and then use it to form an equation for the price $P$ of one quesadillas.

c Follow the steps used in Part A or use the found expressions for the average cost per quesadilla and the price of one quesadilla from Part B.

d Remember that the chef has a fixed weekly cost of $$75.$ Then, think about how that cost is distributed between quesadillas depending on the number of quesadillas sold.

a First, by substituting $25$ for $x,$ the total cost of making $25$ quesadillas can be found.

$C(x)=0.7x+75$

Substitute

$x=25$

$C(25)=0.7(25)+75$

Multiply

Multiply

$C(25)=17.5+75$

AddTerms

Add terms

$C(25)=92.5$

$Cost per Quesadilla:2592.5 =$3.70 $

It is given that the chef wants to earn $$0.30$ from each quesadilla. By adding that value to the cost of making one quesadilla, the price of one quesadilla can be determined.
$Price of a Quesadilla:3.70+0.30=$4.00 $

b To determine how many quesadillas the chef should sell for the price of one quesadilla to be $$1.50,$ the function for the price of one quesadilla should be found. First, by dividing the total cost of making quesadillas by the number of quesadillas $x,$ the average cost of one quesadilla can be calculated.

$Average Cost per Quesadilla:x0.7x+75 $

Next, add $$0.30.$ This is the profit the chef wants to earn from each quesadilla sold. In doing this, the ideal sales price $P$ of making one quesadilla can be found.
$P(x)=x0.7x+75 +0.3 $

Finally, to find the number of quesadillas the chef should sell, substitute $P$ with $$1.5$ and solve the equation for $x.$
Therefore, the chef should sell $150$ quesadillas each week for the price of one quesadilla to be $$1.50.$
c The total cost of making $10$ quesadillas can be found by substituting $10$ for $x$ into the given function $C(x).$

$C(x)=0.7x+75$

Substitute

$x=10$

$C(10)=0.7(10)+75$

Multiply

Multiply

$C(10)=7+75$

AddTerms

Add terms

$C(10)=82$

$Cost per Quesadilla:1082 =$8.2 $

To find the sale price of one quesadilla, add $$0.30$ to the cost of one quesadilla. Also, the function $P(x)$ from Part B can be used.
$Price of a Quesadilla:8.2+0.30=$8.50 $

Similarly, the total cost, cost per quesadilla, and sales price per quesadilla can be calculated for the rest of the given numbers of sold quesadillas. Number of Quesadillas Sold | $10$ | $25$ | $50$ | $75$ | $100$ | $200$ |
---|---|---|---|---|---|---|

Total Cost | $$82$ | $$92.50$ | $$110$ | $$127.50$ | $$145$ | $$215$ |

Cost per Quesadilla | $$8.20$ | $$3.70$ | $$2.20$ | $$1.70$ | $$1.45$ | $$1.08$ |

Sales Price per Quesadilla | $$8.50$ | $$4.00$ | $$2.50$ | $$2.00$ | $$1.75$ | $$1.38$ |

d The price of one quesadilla gets smaller as the number of sold quesadillas increases because there is a fixed weekly cost of adding a new dish to the menu. As more quesadillas are sold, this fixed cost is shared by more quesadillas. As a result, the average cost and, consequently, the price per quesadilla gets smaller.

Example

Let $M(x,y)$ and $N(z,w)$ be two different points.
### Answer

### Hint

### Solution

a Is there always a linear function whose graphs goes through $M$ and $N?$ Please explain.

b Find a linear equation in the standard form such that $(2,5)$ and $(-3,1)$ are the solutions to the equation.

c Find a linear equation in the standard form such that $(4,-2)$ and $(4,4)$ are the solutions to the equation.

a No, if the $x-$coordinates of the points are the same.

b **Example Solution:** $x−2.5y=-8$

c **Example Solution:** $x=4$

a Consider the case where the $x-$coordinates of $M$ and $N$ are the same. Use the definition of a function.

c Examine the coordinates of the given points. What can be said about the line on which they lie?

a Recall that a function is a relation in which each input is assigned to exactly one output. Now, consider the situation where $M$ and $N$ have the same $x-$coordinates.

$M(3,y)andN(3,w) $

In such a case, for the input $x=3,$ the function will have two different outputs, $y$ and $w.$ This contradicts the definition of a function. Therefore, there is not always a function whose graph passes $M$ and $N.$ However, if the points have different $x-$coordinates, there is always a function whose graph contains those points. b Start by reviewing the standard form of an equation.

$Ax+By=C $

Next, substitute the coordinates of the two given points, $(2,4)$ and $(-3,2),$ to obtain two equations.
$2A+4B=C(I)-3A+2B=C(II) $

To eliminate one variable, for example $B,$ multiply the second equation by $2$ so that the coefficients before $B$ match in both equations.
$-3A-6A +2B=C∣∣∣ ×2⇕+4B=2C $

Now, subtract the obtained equation from Equation $(I).$
$2A+4B_{−}-6A+4B8A+4B =C=2C=-C $

There is not enough information to solve for both $A$ and $C.$ Therefore, choose a convenient value of $C,$ and then use it to solve the equations for $A$ and $B.$ For example, let $C$ be equal to $-8.$ Use this value to calculate $A.$
Now, the only unknown value is $B.$
$A=1B=?C=-8 $

By substituting the known values of $A</$