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Pre-Algebra View details
3. Linear Functions
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eCourses /
Pre-Algebra /
Chapter 8
3. 

Linear Functions

Student Learning Objectives:
  • Identify functions by the shape of their graph
  • Classify data values as quantitative and categorical, discrete and continuous

Why
Linear functions are essential in understanding relationships between variables and their real-world applications. This lesson delves into how numerical data is represented through linear relationships and explains the differences between continuous and discrete quantities. It also highlights the importance of domains, distinguishing between continuous and discrete scenarios. By mastering these concepts, students gain a deeper understanding of how linear functions apply to modeling and analyzing data in areas like economics, science, and technology. These skills build a strong foundation for exploring more advanced mathematical concepts and solving real-life problems effectively.

Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
15 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Linear Functions
Slide of 15
This lesson explores linear functions by reviewing concepts such as slope and functions. Following that will be a discussion about the classification of variables. Additionally, real-world problems will be solved to provide a comprehensive understanding of such functions.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

Using Linear Functions to Analyze Computer Temperatures

Vincenzo and Magdalena are two friends eager to explore the world of gaming and programming. They are developing their own mobile game by coding and testing together. As part of their development process, they are monitoring the temperature of their computers. The diagram shows how the temperature of each changes over time.

a Find and interpret the slopes and the y-intercepts of the lines.
b For each graph, write a linear function that relates y to x.
Discussion

Linear Function

A linear function is a function with a constant rate of change. A linear function can be represented by a linear equation in two variables. Graphically, a linear function is a nonvertical line.

Using that line, the rate of change can be determined by finding the horizontal change Δ x and the vertical change Δ y between any two points on the line. Any function whose graph is not a straight line cannot be linear.
Discussion

Nonlinear Function

A nonlinear function is a function that does not have a constant rate of change. In other words, a nonlinear function is a function whose graph is not a straight line. Instead, the graph may take the form of any curve other than a straight line.

The functions above are all examples of nonlinear functions since their rate of change vary.
Example

Randomly Placed Characters in The Game

Vincenzo and Magdalena have planned for two characters in their upcoming mobile game. When the game begins, these characters will be positioned at random points within the game environment. Let the points where these characters are located be M(x,y) and N(z,w)

Graph that locates points M and N on a coordinate plane
Is every line that passes through points M and N necessarily a linear function?

Hint
What is the condition for being a function? What happens if the x-coordinates of M and N are the same.

Solution
Recall that a function is a relation in which each input is assigned to exactly one output. Now, consider the case where M and N have the same x-coordinate, for example 3. Point & Coordinates M & ( 3,y) N & ( 3,w) In such a case, when the input is set to 3, there will be two separate outputs, namely y and w. This contradicts the definition of a function. Therefore, a line that passes through points with the same x-coordinate is not a function.

A function whose graphs passes through M and N

However, if the points have different x-coordinates, there is always a function through those points.

Pop Quiz

Identifying Functions From Graphs

Analyze the given graph and determine whether it represents a function. If it is a function then determine if it is a linear function or a nonlinear function.

random relations
Discussion

Numerical Data

Numerical data, also called quantitative data, is measurable data expressed using numbers. Some examples of numerical data are age, height, and weight of a person. Consider the following personal information.

Person Age Weight Shoe Size Number of Siblings Eye Color
Dejen 15 43.4 6 0 Blue
Madge 16 56.7 7 3 Green
Yulia 20 64.8 7.5 1 Brown

A person's age, weight, shoe size, and number of siblings are examples of numerical data. In contrast, a person's eye color cannot be expressed in numbers and is an example of categorical data. Additionally, numerical data can be either discrete data that can only take specific values or continuous data that can take any value within an interval.

Data Numerical? Type
Age Yes ✓ Continous
Weight Yes ✓ Continous
Shoe Size Yes ✓ Discrete
Number of Siblings Yes ✓ Discrete
Eye Color No * Non-applicable
Discussion

Continuous Quantity

A quantity that can take any value within a given interval is a continuous quantity. Such quantities are not limited to specific, separate values, and they can be measured very precisely. Length, time, temperature, weight, and age are examples of continuous quantities.

Here, each point on the number line corresponds to a specific age, and the line itself represents the possible ages. A person's age can be any real number, depending on the precision of the measurement. In using whole years, someone's age might be said to be 5 years old. With more precise measurements, age could be expressed as 5 and a 12 years, 8 years and 2 months, or an age like 8 years and 88 days.
Discussion

Discrete Quantity

A discrete quantity is a quantity that can only take distinct, separate values in an interval. There are no values between these distinct values. The number of students in a class, the number of tickets for a concert, and the number of goals scored in a soccer match are examples of discrete quantities. These values are countable and typically represented by integers or whole numbers.

Note that the value of the quantity can only be specific amounts such as 0, 1, or 2, as a fraction or part of a discrete quantity is not an option. Imagine buying only part of a concert ticket! Now, although discrete quantities are often restricted to whole numbers, there are exceptions. Depending on the context, discrete quantities can take values from a set like {7,7.5,8,8.5,9}.

Extra

 Exceptions Explained
Shoe sizes, for example, can include half-sizes such as 7, 7.5, 8, 8.5, and 9. Shoe Sizes { 7,7.5,8,8.5,9} This provides more options for people to find a better fit. Another example can be a grading system where students can receive scores that include half-points. Scores in a Specific Grading System { 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5} The scores are still discrete because they are specific and separate values. Each score is distinct from the others and there are no values between them. Therefore, in the context of shoe sizes and grading systems, discrete quantities can indeed be fractions or decimal numbers.
Example

Identifying Variables in Game Development

In the process of developing their mobile game, Vincenzo and Magdalena find it necessary to analyze certain numerical data related to user interactions within the game. After some brainstorming, they identify several variables for consideration. L: & the number of levels in the game T: & the time it takes for a player to complete a level S: & the score of a player V: & the speed at which the character moves Classify these variables as either continuous or discrete.

Hint
A continuous quantity can take any value in a certain interval. A discrete quantity can only take distinct values, so there is a gap between possible values.

Solution
To determine if a variable is continuous or discrete, check if there is a gap between possible values. Analyze each given variable separately to reach an answer.

Variable L

The number of levels is typically counted in whole numbers. It cannot take a real number value, so it is discrete.

Variable T

The time it takes for a player to complete a level can be any real number. Time can be measured with high precision, including fractions of seconds. Therefore, T is continuous.

Variable S

A player's score can typically take whole numbers. Scores are limited to whole numbers, so it is discrete.

Variable V

The speed at which the character moves can be measured on a continuous scale. That is, it can take any real value, including fractions. Therefore, V is continuous.

Summary

In conclusion, discrete variables take on distinct values, often whole numbers. Continuous variables, on the other hand, can take any value within a given interval.

Variable Can it take all values in a certain interval? Type
L No Discrete
T Yes Continuous
S No Discrete
V Yes Continuous
Discussion

Continuous Domain

A function is said to have a continuous domain when the independent variable of the function is a continuous quantity. The graph of such a function is typically a curve or a line. Consider the graph of the function, which shows the relationship between a person's age and weight.

The domain of this function consist of all real numbers from 0 to 10, indicating that the function has a continuous domain.
Discussion

Discrete Domain

A function is said to have a discrete domain when the independent variable of the function is a discrete quantity. A function of this type can be identified by its graph, which is composed of any number of unconnected points. Consider the function that models the costs of concert tickets, specifically for 0, 1, 2, 3, and 4 tickets.

The domain of this function is the set of integers from 0 to 4, indicating that the function has a discrete domain.
Example

Modeling In-Game Spending with Linear Functions

Vincenzo and Magdalena are continuing the development of their own mobile game. In their game, players start with 100 virtual coins and spend 5 coins per day on in-game activities.

a Write a function to represent the amount of coins y a player has left x days after starting the game.
b Make a table to find the remaining number of coins for a player after 2, 4, 6, 8, and 10 days from starting the game.
c Graph the function. Does the function have a continuous domain or a discrete domain?

Answer
a Function: y = - 5x+100
b Table:
Days, x Number of coins, y
2 90
4 80
6 70
8 60
10 50
c Graph:

The function have a discrete domain.

Hint
a The number of coins a player has can be expressed as a linear function. Use the slope-intercept form to write the function.
b Substitute the number of days into the function from Part A.
c Draw a coordinate plane where the horizontal axis represents the number of days and the vertical axis represents the remaining number of coins.

Solution
a Recall that a player starts with 100 coins and they can spend 5 coins each day. This refers to a decrease at a constant rate. Therefore, the number of coins a player has (y) can be expressed as a linear function of the number of days the game has been played (x). The slope-intercept form can be used to write that function.

y = mx + b In this form, m is the slope and b is the y-intercept. In this case, the initial amount of virtual coins is 100, so b = 100. The spending rate is 5 coins per day, so the slope is - 5 because the coins are decreasing. Therefore, the linear function is as follows. y = - 5x + 100

b Make a table of values for the function written in Part A.
Days x - 5x+100 Number of Coins y
2
4
6
8
10

How about finding the remaining number of coins for a player after 2, 4, 6, 8, and 10 days from starting the game? Try substituting the values of days into the formula and evaluate.

Days x - 5x+100 Number of Coins y
2 - 5( 2) + 100 90
4 - 5( 4) + 100 80
6 - 5( 6) + 100 70
8 - 5( 8) + 100 60
10 - 5( 10) + 100 50

Great work so far!

c The function can be graphed using the table created in Part B.
Days x Number of Coins y
2 90
4 80
6 70
8 60
10 50

Plot the ordered pairs (x,y) on a coordinate plane where the x-axis represents the number of days and the y-axis represents the remaining number of coins.

Since the function has a discrete domain, the points are not connected with a straight line. There are missing points in this graph. When other points are included, the graph of the function will be as follows.

Example

Modeling Battery Consumption with Linear Functions

As part of their development process, Vincenzo and Magdalena monitor the battery power consumption of a test laptop.

The table shows the percentage, y in decimal form, of battery power remaining x hours after the laptop is turned on.

Hours, x 0 2 3
Power Remaining, y 1.0 0.4 0.1
a Write and graph a linear function in point-slope form that relates the percentage of power remaining y to the number of hours x.
b Interpret the slope, x-intercept, and y-intercept in this context.
c After how many hours is the battery power at 25 %?

Answer
a Function: y -0.1 = - 0.3(x-3)

Graph:

b See solution.
c 2.5 hours

Hint
a First, use the given points to calculate the slope. Then, identify the y-intercept.
b Recall the meaning of x and y in the context of the problem.
c Use the function obtained in Part A.

Solution
a Take a look at the given table. It is asked to write a linear function that relates y to x in point-slope form.
Hours, x 0 2 3
Power Remaining, y 1.0 0.4 0.1

Recall that functions written in point-slope form follow a specific format. y - y_1 = m(x- x_1) In this form, m is the slope and ( x_1, y_1) is a point on the line. Start by identifying the slope using two points from the given table. Substitute, for example, the points (0,1) and (2,0.4) into the slope formula and calculate m.

m = y_2-y_1/x_2-x_1
SubstitutePoints

Substitute ( 0,1) & ( 2,0.4)

m=0.4- 1/2- 0
▼
Simplify right-hand side
SubTerms

Subtract terms

m=- 0.6/2
CalcQuot

Calculate quotient

m=- 0.3

The slope of the line is - 0.3. Considering the given table, the function passes through the point ( 3, 0.1). Now that the slope and a point is known, the function can be written. y - 0.1 = - 0.3(x- 3) Finally, plot the given points on a coordinate plane and draw a line that passes through them. The graph of the function is a solid line because the function has a continuous domain.

b Using the graph drawn in Part A, the slope and intercepts of the line can be determined. The line intersects the y-axis at (0,1), while it intersects the x-axis at (3 13,0) because each unit on the x-axis is 13 in length. To confirm it, substitute 0 for y in the equation of the line and solve the equation for x.

y -0.1 = - 0.3(x-3)
Substitute

y= 0

0-0.1= - 0.3(x-3)
▼
Solve for x
SubTerm

Subtract term

- 0.1= - 0.3(x-3)
DivEqn

.LHS /(- 0.3).=.RHS /(- 0.3).

- 0.1/- 0.3= x-3
ExpandFrac

a/b=a * (- 10)/b * (- 10)

1/3= x-3
AddFrac

Add fractions

3 +1/3 = x
Rewrite

Rewrite 3 +1/3 as 3 13

3 13 = x
RearrangeEqn

Rearrange equation

x = 3 13

The line does, in fact, intersect the x-axis at (3 13,0)

As found in Part A and shown on the above graph, the slope of the line is -0.3. It is negative because the power remaining decreases in time. 0.3=30/100=30 % Therefore, the slope indicates that the power decreases by 30 % per hour. Note that the y-intercept is (0,1). 1=100 % The y-intercept indicates that the battery power is at 100 % when the laptop is turned on. The x-intercept is (3 13,0). It indicates that the battery lasts 3 13 hours, which can also be rewritten as follows.

3 13h
▼
Rewrite
Rewrite

Rewrite 3 13h as 3h + 1/3h

3h + 1/3h
HourToMin

1h=60 min

3h + 1/3 * 60min
MoveRightFacToNumOne

1/b* a = a/b

3h + 60min/3
CalcQuot

Calculate quotient

3h + 20min
Rewrite

Rewrite 3h + 20min as 3h and20min

3h and20min

The battery lasts 3 hours and 20 minutes.

c The function rule can be used to find the number of hours it will take to get 25 percent of battery power.

y -0.1 = - 0.3(x-3) Since the y-variable represents the power remaining in decimal form, 25 % must be written as a decimal first. 25 % = 0.25 Now substitute y= 0.25 into the function and solve it for x.

y -0.1 = - 0.3(x-3)
Substitute

y= 0.25

0.25 -0.1 = - 0.3(x-3)
▼
Solve for x
SubTerm

Subtract term

0.15 = - 0.3(x-3)
DivEqn

.LHS /(- 0.3).=.RHS /(- 0.3).

0.15/- 0.3= x - 3
CalcQuot

Calculate quotient

- 0.5 = x-3
AddEqn

LHS+3=RHS+3

2.5 = x
RearrangeEqn

Rearrange equation

x=2.5

Therefore, after 2.5 hours the battery power will be at 25 percent.

Closure

Making Conclusions About Two Linear Functions

Finally, the challenge presented at the beginning of the lesson will be solved. In the challenge, Vincenzo and Magdalena are monitoring the temperatures of their computers which tend to increase while running applications.

It is helpful to answer the questions by recalling the concepts and rules presented throughout the lesson.

a Find and interpret the y-intercept and the slope of each line.
b For each graph, write a linear function that relates y to x.

Answer
a Table:
Vincenzo's Computer Magdalena's Computer
y-intercept 32 24
The initial temperature of the computer is 32^(∘)C. The initial temperature of the computer is 24^(∘)C.
Slope 6 8
Its temperature increases by 6^(∘)C per hour. Its temperature increases by 8^(∘)C per hour.
b Vincenzo's Computer: y = 6x+32
Magdalena's Computer: y = 8x+24

Hint
a Identify the point where each line intersect the y-axis. For each line, identify two points on the line and substitute them into the Slope Formula to find its slope.
b Use the slope-intercept form to write the functions.

Solution
a The y-intercept of a line is the y-value where the line crosses the y-axis.

The y-intercept of the blue line is 32 and the y-intercept of the red line is 24. Note that the y-axis represents the temperature in Celsius degrees. Since the time is zero at those points, the y-intercepts represent the initial temperatures of the computers.

Vincenzo's Computer Magdalena's Computer
y-intercept 32 24
The initial temperature of the computer is 32^(∘)C. The initial temperature of the computer is 24^(∘)C.

Now the slope of each line will be found. The Slope Formula can be used to find the slopes. m = y_2-y_1/x_2-x_1 It is necessary to determine one more point on each line in order to use the formula.

The blue line passes through the points (0,32) and (4,56), while the red line passes through the points (0,24) and (1,32). Substitute these points into the Slope Formula.

Blue Line Red Line
Points ( 0, 32) and ( 4, 56) ( 0, 24) and ( 1, 32)
m = y_2-y_1/x_2-x_1 56- 32/4- 0 32- 24/1- 0
Evaluate 24/4 = 6 8/1 = 8

In this context, the slope of the functions indicate how quickly the temperature of a computer rises within an hour.

Vincenzo's Computer Magdalena's Computer
Slope 6 8
Its temperature increases by 6^(∘)C per hour. Its temperature increases by 8^(∘)C per hour.

Magdalena's computer temperature increases faster than Vincenzo's when both are running the applications. Note also that both functions have continuous domains as the inputs can take any value.

b The slope and the y-intercept of the lines are found in Part A.
Blue Line Red Line
y-intercept 32 24
Slope 6 8

Using these values, the equation of each line can be written in slope-intercept form. y= mx+ b In this form, m is the slope and b is the y-intercept.

Blue Line Red Line
y-intercept 32 24
Slope 6 8
Equation y= 6x+ 32 y= 8x+ 24

The blue line corresponds to Vincenzo's computer temperature, whereas the red line corresponds to Magdalena's computer temperature.



Level 1
Level 2
Level 3
Linear Functions
Exercise 2.1
Tadeo is saving money for a vacation. He already has $300. He plans to save an additional $75 per month. Write a function in slope-intercept form that represents his savings y for any number of months x.

What is the graph of the function?

We are asked to write a linear function that represents Tadeo's savings y in dollars for any number of months x. We need to write it in slope-intercept form. y= mx+ b For an equation in this form, m is the slope and b is the y-intercept. We will find the slope and the y-intercept one at time. Let's start by recalling that the definition of the slope of a linear function can be written in terms of rise and run. m=rise/run In this case, his savings increase by $ 75 for each month. Therefore, the rise, or change in y, is 75 and the run, or change in x, is 1. We can substitute these values into the formula for the slope to calculate m. m=75/1= 75 Next, we will find the y-intercept which is the amount of money at the beginning. Since Tadeo has already $300, we can say that b= 300. Now that we have the slope and the y-intercept we can write our final equation. y= 75x+ 300


We will graph the function we wrote in Part A. Tadeo's Savings y = 75x + 300 First we will determine the domain of the function. In this context, the independent variable, representing the number of months, is restricted to whole numbers. This restriction stems from the lack of information about the regularity of her savings — whether they occur consistently on a weekly or daily basis. Whole numbers ensure a meaningful representation of Tadeo's savings over discrete months. Tadeo's Savings: & y = 75x + 300 Domain: & Whole numbers The function has a discrete domain. Now we substitute some whole number values for x and find the corresponding values of y. These x- and y-values will create a set of ordered pairs.

x 75x-300 y (x,y)
0 75( 0)+300 300 ( 0, 300)
1 75( 1)+300 375 ( 1, 375)
2 75( 2)+300 450 ( 2, 450)
3 75( 3)+300 525 ( 3, 525)
4 75( 4)+300 600 ( 4, 600)

Now we will graph the ordered pairs from the table. We will not connect the points with a line because its domain is discrete.

This matches the graph in C.

E
Hint & Answer
S
Solution
Store A offers packages with comic books with a poster. All comic books in a pack have the same price, and each poster costs the same.

1 poster and 4 comics for $10.50; 1 poster and 12 comics for $16.50
 Write a linear function in the slope-intercept form that represents the cost y of a package containing any number of comic books x.

Store B sells a similar package, charging the same price for each comic as Store A and selling the poster for $8.50. Which store has the better deal?

We are asked to write a linear function that represents the cost y of a package containing any number of comic books x. It will be in the slope-intercept form. y= mx+ b For an equation in this form, m is the slope and b is the y-intercept. We need the slope and the y-intercept of our function. We can do this by using any two points lying on the function's graph. Let's identify two points from the given information. A poster is given in each scenario, so we can consider the cost of a poster as a constant, which we will address later.

Given Information Point (x,y)
1 poster and 4 comics cost $ 10.50. ( 4, 10.50)
1 poster and 12 comics cost $ 16.50. ( 12, 16.50)

Now we can use the obtained points to calculate the slope of our function, m. We will start by substituting the points into the Slope Formula.

In our function, a slope of 0.75 means that if we buy 1 more comic book, we will pay $ 0.75 more. Now that we know the slope, we can write a partial version of the equation. y= 0.75x+ b To complete the equation we also need to determine the y-intercept, b. Since we know that the given points will satisfy the equation, we can substitute one of them into the equation to solve for b. Let's use ( 4, 10.50).

A y-intercept of 7.50 means that a poster costs $ 7.50. We can now complete the equation. y=0.75x+ 7.50 ⇔ y = 0.75x+7.5

We are given that Store B has the same comic book price as Store A, and the poster costs $8.50. We want to determine which store has the better deal. Let's start by recalling the equation of the linear function obtained in Part A. Store A y = 0.75x+ 7.5 For Store A, the price per comic book is $ 0.75, so is the price in Store B. However, the price of a poster is $ 8.50. Therefore, the function that models the cost of a package containing x number of comics will be as follows. Store B y =0.75x+ 8.5 Note that at both stores the price of each comic book is the same, but in the second store the poster costs $ 8.50, which is greater than $ 7.50. 8.50 > 7.50 Since each package contains the poster, it will always cost more in Store B. This means that Store A has the better deal.

E
Hint & Answer
S
Solution

Why does a continuous linear function have an infinite number of solutions?

A linear function is a function of the following form. y=mx+b As the name suggests, the graph of a linear function is a line.

The domain of a linear function is all real numbers unless otherwise stated. Domain of a Linear Function - ∞ < x < ∞ The solution to a linear function is an ordered pair ( x, y) that makes the function equation true. y=m x+b There are infinitely many real numbers in its domain. This means that there are infinitely many x-values, and corresponding y-values, that satisfy the equation.

As a result, there is always an infinite number of solutions to the equation. Linear Function: & y=m x+b Number of Solutions:& Infinitely many We can also think of it in a different way. A linear function consists of infinitely many points. Each of these points is a solution. Then there are infinitely many solutions to the function. Therefore, the answer is B.

E
Hint & Answer
S
Solution

The cost to rent a bike from two different companies is shown, where x is the time in hours and y is the total cost.

If you plan to rent a bike for 7 hours, which company would be the more economical choice?

We are given costs from two different companies to rent a bike. We are asked to determine which company should we use if we want to rent a bike for 7 hours. First, let's analyze the cost Company A offers.

Company A

The costs of renting a bike from this company are represented with a graph. Let's find its slope and y-intercept so that we can write the equation of the graph.

We can see that this graph passes through the origin, and therefore the y-intercept is equal to 0. The slope of the line is 4, which is the rate of change between the points ( 2, 8) and ( 3, 12). m=12- 8/3- 2 = 4 Now we can write an equation in slope-intercept form which will relate the total cost y for renting a bike and the time in hours x.

Slope-Intercept Form: y=mx+b
y= 4x+ 0 y = 4x

Now to find the cost for renting a bike for 7 hours, we will substitute x=7 into this equation.

We found that it costs 28 dollars to rent a bike for 7 hours for Company A. Now let's check out the other company.

Company B

We are given the costs for renting bikes from Company B in a table. Let's try to find a pattern in this table.

We can see that as the rent time increases by 1 hour, total cost increases by 3 dollars. If we were to plot the points in this table, they would form a line with a slope of 3. Note also that if we subtract 3 from 9, we get 6, which we can interpret it as a fixed cost or initial fee. Then, the following equation represents the table. y = 3x+ 6 We can use this function to calculate the cost for renting a bike for 7 hours.

We found that it costs 27 dollars to rent a bike from Company B for 7 hour, which is less than the cost of renting a bike in Company A. 7Hours Rental Cost ccc CompanyA && CompanyB 28 & > & 27 Therefore, it is more economical to choose Company B to rent a bike for 7 hours.

E
Hint & Answer
S
Solution

Linear Functions

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