Exploring Linear Functions in Algebra

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Analyzing Two Linear Functions

Comparing Rates of Changes

Using Function Notation to Model a Problem

Identifying the Domain and Range of a Function

Using a Function and Inverse Function to Convert Temperatures

Applying a Linear Function to a Real-Life Problem

Finding a Linear Function Given Two Points

Making Conclusions About Two Linear Functions

Catch-Up and Review

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Catch-Up and Review

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Solution

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.

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

Function
Linear Function
Function Notation
Rate of Change
Inverse of Linear Functions
Standard Form of a Linear Function

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.

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The following two linear functions describe the way the girls walk to school separately.

Here,

denotes the time the girls are walking, while

and

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

and

on the coordinate plane.

b Locate

and

on the coordinate plane. Then, interpret those values in terms of the given situation.

c Consider the points with the

coordinates of

on the functions

and

where

is their point of intersection. What is the difference between the

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
	F
	F
	F

When was the temperature decreasing the fastest: between and or between and

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.

In this formula,

represents the change in the output variable and

the change in the input variable. Because the change of temperature

over the change of time

needs to be found, the formula for the rate of change can be rewritten in terms of

and

To apply this formula, the changes in time and temperature will be calculated.

Time Interval
	hours
	hours

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

Time Interval			Rate of Change
	hours		per hour
	hours		per hour

As can be seen, the temperature decreased at an average rate of in the first time interval and in the second time interval. Therefore, the temperature decreased the fastest between and

Example

Along the walk to school, Dominika passes her father's cafe. Running the cafe requires close attention to customer behavior. Her father notices that it is typical to serve customer each night from till Then, starting from the cafe slowly begins to fill with regulars.

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

a Is

a function of

Explain your answer.

b Is

a function of

If yes, consider a function

and interpret the meaning of

in the context of the given situation.

c Mr. Beckett noticed that the number of customers usually increases by an average of

people during the lunch hours from

Which of the following equations best represents that observation?

a No, because the input value of

is associated with three output values

b Yes,

represents the number of customers in the cafe at

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

b To interpret the meaning of

think about what variables

and

denote.

c Analyze the meaning of each given equation. How can the number of customers at

and

be written using a function notation?

a To determine whether

is a function of

recall the definition of a function.

A function is a relation in which each input is assigned to exactly one output.

From

till

there is usually

customer in the cafe. If

is a function of

written as

then the values of

are inputs and the values of

are outputs. Here, the input is

the customer, and the outputs are the hours

and

labeled

and

respectively. This relation can be visualized with a mapping diagram.

In the diagram, it can be observed that there would be more than one output

associated with the single input value of

This fact contradicts the definition of a function. Therefore,

cannot be a function of

b Contrary to Part A, if

is a function of

each input

is assigned to exactly one output

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 more than one output. Therefore,

is a function of

To interpret the meaning of

recall that the input

represents the time in hours since midnight, while the output

represents the number of people in the cafe.

The expression

represents the corresponding value of

when

In this situation,

represents

and

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
	At there are customers in the cafe.
	The difference between the number of customers at and the number of customers at is people.
	The average change of the number of customers per hour between and is people.
	The number of people at the cafe equals the number of customers at plus more customers.

The second equation can be rephrased as the number of customers at increased by since 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 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 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 who attended it.

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

b What is the range of this function if

is the amount of money earned from the game? Write the answer as a compound inequality.

c Write the function for the average profit

from the game if the cost of organizing the game is

a Analyze what minimum and maximum values

can have.

b The range is the set of all the outputs of the function.

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

who will attend the game. Let

be the amount of money earned from the game. Now, the described relation can be written as follows.

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

Since

represents the number of people attending the game, it can only have non-negative values.

Also, Soldier Field is said to have a maximum capacity of

Therefore,

cannot have values greater than

Additionally, since

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.

b The range of a function is the set of all the outputs of the function. In this case, the outputs consist of all

values, representing the amount of money earned from selling tickets. If nobody attended the game — meaning no tickets were sold — the earnings would be

This is the minimum value that

could have.

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

and the stadium's total capacity is

so the product of these values will give the maximum possible earnings.

Finally, because both the price of a ticket and the number of people are integers,

will also be an integer number. With this information, the range of the function can now be summarized.

c Because the cost of organizing the game is

the profit

can be determined by subtracting this quantity from the money earned by selling the tickets. Note that the money

earned by selling the tickets is equal to the product of the price of a ticket and the number of people

that will attend the game.

Now, the cost of organizing the game can be subtracted from the money earned to get the profit.

It can be seen that

is a function of

This can be explicitly indicated by writing the equation for the average profit in function notation.

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.

Dominica's website that converts two temperatures from Kelvins to degrees Fahrenheit

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
The body temperature in Kelvins is while in degrees Fahrenheit, it is

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

a Write a linear function

that assigns to a temperature in Kelvins its equivalent in degrees Fahrenheit.

b Find the inverse of the function

and explain its meaning in terms of temperature conversions.

c Is there a temperature that is the same in Kelvins and degrees Fahrenheit? If yes, round the answer to one decimal place.

a Since

is a linear function, it has the form of

Use the given information about the zero point and body temperature to find the values of

and

b To find the inverse of a function, switch the

and

places, and then solve the equation for

c If there is a temperature which is the same in Kelvins and degrees Fahrenheit, then

would be equal to

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.

For this situation, the inputs

will represent a temperature in Kelvins and the outputs

will represent the corresponding temperature in degrees Fahrenheit. It is known that

Kelvin equals

Therefore, the value of the constant

can be calculated by substituting

for

and

for

SubstituteII

▼

Solve for

ZeroPropMult

Zero Property of Multiplication

IdPropAdd

Identity Property of Addition

RearrangeEqn

Rearrange equation

Now that the value of

is found, substitute it into the above formula of a linear function.

Using a similar reasoning, the given body temperatures in Kelvin and degrees Fahrenheit can be used to find the value of

This time,

will be substituted for

and

for

into the partial function found previously.

SubstituteII

▼

Solve for

AddEqn

RearrangeEqn

Rearrange equation

DivEqn

Finally, the function that converts Kelvins into degrees Fahrenheit can be completed.

b In order to find the inverse of the function

first replace

with

describes the input-output relationship of the function.

Next, because the inverse of a function reverses

and

switch

and

places in the function rule.

Now, the obtained equation should be solved for

▼

Solve for

Rewrite as

Rearrange equation

Therefore, the inverse function of

has been found. In the context of this situation, this inverse function converts degrees Fahrenheit into Kelvins.

Dominika can now use these two functions to implement the desired feature on her amazing app.

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c The function found in Part A can now be used to determine a temperature whose values in Kelvins and in degrees Fahrenheit are the same. If such a value exists, the output of the function would be the same as the input.

Therefore, substitute

with

and solve the equation using inverse operations.

Substitute

▼

Solve for

SubEqn

DivEqn

RoundDec

Round to decimal place(s)

It can be concluded that

Kelvins equals

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!

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Her father has calculated that there would be a weekly fixed cost of

for adding a new dish plus an additional

for the ingredients per quesadilla. The total cost

for preparing

quesadillas can be represented by a linear function.

Now, the chef needs to figure out how much to charge for one quesadilla. Ideally, he wants to earn

per each quesadilla sold.

a If the chef sells

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

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.

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

a Cost per Quesadilla:

Price of One Quesadilla:

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

quesadillas and then add

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

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

Then, think about how that cost is distributed between quesadillas depending on the number of quesadillas sold.

a First, by substituting

for

the total cost of making

quesadillas can be found.

Substitute

Multiply

AddTerms

Add terms

Next, divide the total cost by

to calculate the cost of making one quesadilla.

It is given that the chef wants to earn

from each quesadilla. By adding that value to the cost of making one quesadilla, the price of one quesadilla can be determined.

b To determine how many quesadillas the chef should sell for the price of one quesadilla to be

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

the average cost of one quesadilla can be calculated.

Next, add

This is the profit the chef wants to earn from each quesadilla sold. In doing this, the ideal sales price

of making one quesadilla can be found.

Finally, to find the number of quesadillas the chef should sell, substitute

with

and solve the equation for

Substitute

▼

Solve for

Therefore, the chef should sell

quesadillas each week for the price of one quesadilla to be

c The total cost of making

quesadillas can be found by substituting

for

into the given function

Substitute

Multiply

AddTerms

Add terms

The cost per quesadilla can be calculated either by dividing the found total cost by

or by substituting

with

into the expression for the average cost of a quesadilla found in Part B.

To find the sale price of one quesadilla, add

to the cost of one quesadilla. Also, the function

from Part B can be used.

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.

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

and

be two different points.

Graph that locates points M and N on a coordinate plane

a Is there always a linear function whose graphs goes through

and

Please explain.

b Find a linear equation in the standard form such that

and

are the solutions to the equation.

c Find a linear equation in the standard form such that

and

are the solutions to the equation.

a No, if the

coordinates of the points are the same.

b Example Solution:

c Example Solution:

a Consider the case where the

coordinates of

and

are the same. Use the definition of a function.

b Since there is not enough information to solve the equations for all three unknown variables, choose a convenient value of

then solve equations for

and

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

and

have the same

coordinates.

In such a case, for the input

the function will have two different outputs,

and

This contradicts the definition of a function. Therefore, there is not always a function whose graph passes

and

However, if the points have different

coordinates, there is always a function whose graph contains those points.

A function whose graphs passes through M and N

b Start by reviewing the standard form of an equation.

Next, substitute the coordinates of the two given points,

and

to obtain two equations.

To eliminate one variable, for example

multiply the second equation by

so that the coefficients before

match in both equations.

Now, subtract the obtained equation from Equation

There is not enough information to solve for both

and

Therefore, choose a convenient value of

and then use it to solve the equations for

and

For example, let

be equal to

Use this value to calculate

Substitute

NegNeg

DivEqn

Now, the only unknown value is

By substituting the known values of

and

into either Equation

the value of

can be calculated.

SubstituteII

▼

Solve for

MultByOne

SubEqn

DivEqn

Finally, the equation can be completed.

Note that if another value of

was chosen, the coefficients of

and

would be different. For instance, if

then

and

Then, a different equation is obtained.

However, these are equivalent equations of the same line, as the first equation multiplied by

is the same as the second equation.

If those equations were rewritten in slope-intercept form, they would be the same. This observation strongly emphasizes an advantage of the slope-intercept form: for every line, there is only one equation that describes it.

c Start by analyzing the coordinates of the given points.

As can be noticed, the points have the same

coordinates. Therefore, they both lie on the vertical line

Additionally, by multiplying both sides of this equation by some number, an equivalent equation that also contains these points can be found.

Keep in mind that these equations do not describe a function, as for one input of

there are many different outputs.

Closure

Finally, the challenge presented at the beginning will be solved. It stated that Dominika and her friend Zosia live relatively close to each other. Each morning they leave their homes to walk to school, and at some point, they meet and continue the trip together. Usually, when walking separately, they walk at different speeds.

Error loading image.

The following two linear functions describe the way the girls go to school separately.

Here,

denotes the time the girls are walking, while

and

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

and

on the coordinate plane.

b Locate

and

on the coordinate plane. Then, interpret those values in terms of the given situation.

c Consider the points with the

coordinates of

on the functions

and

where

is their point of intersection. What is the difference between the

coordinates of these points?

a Dominika:

Zosia:

Graph:

b The value

represents Dominika's distance before she has begun her walk from her home to school. Likewise,

denotes Zosia's distance from Dominika's when she, too, has yet to begin her walk.

a Think of what the slopes of the functions represent. Remember, steeper lines have greater slopes.

b Remember that the constant in the slope-intercept form represents the

intercept of the function. Recall what

and

denote.

c Identify the slopes of the functions from their equations. Use the slopes to find how much each function will rise when moving

units to the right.

a By examining the given functions, it can be noted that they are given in the slope-intercept form. The slopes of these functions represent the walking speeds of the girls. Since

has a greater slope, this function must correspond to Dominika. Therefore,

corresponds to Zosia.

Note that a function with a greater slope has a steeper graph. On the given diagram, the blue line is steeper than the purple line. With this information, the functions can now be labeled.

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b In the slope-intercept form, a constant represents the

intercept of the function. Therefore,

and

are the

coordinates of the points where the functions intersect the

axis.

It is given that

represents the time spent walking, and

and

represent each girl's distance from Dominika's home. From the diagram, it can be concluded that

equals

This makes sense, since Dominika is going to school from her home, so when she has yet to start her walk, her distance from her house is

In the case of Zosia,

denotes her distance from Dominika's house when she has not started her walk to school. If she is also starting from her own home, then

is the distance between the houses of the two girls.

c To find the difference between the

coordinates of the given two points, the slope of the functions will be used. First, identify the slopes of the functions using their equations.

Recall that the slope gives the number of units a function rises when moving

unit to the right. By using this information, the number of units that each function rises when moving

units to the right can be determined.

Now, by adding this values to

the

coordinates of the points with the

coordinate of

can be found.

Finally, the difference between the found

coordinates can be determined.

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Number of Quesadillas Sold

Total Cost

Cost per Quesadilla

Sales Price per Quesadilla