lesson 12.1.2 version 2.5-student - De Anza Collegenebula2.deanza.edu/~mo/Math217/04-lesson_12.1.2... · statway®"studenthandout"""|"3""" lesson"12.1.2"" linearfunctions!! ©2013!thecarnegie!foundation!forthe!advancement!of!teaching!

© 2013 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 2.5, STATWAY® -‐ STUDENT HANDOUT

STATWAY® STUDENT HANDOUT

Lesson 12.1.2 Linear Functions

INTRODUCTION Jean lives in a housing co-‐operative, where she shares food and housing expenses with several roommates. Jean needs to buy meat for her housing co-‐operative. She has two options:

• She can go to a Fresh-‐Plus store and pay $4.50 per pound. • Or, she can also go to a warehouse store, like Costco or Sam’s Club and pay $3 per pound.

Fresh-‐Plus is near her housing co-‐operative. A trip to Fresh-‐Plus will cost $0.50 for gas. The warehouse store is further away. A trip to the warehouse store will cost $5 for gas. Our goal is to write a mathematical function (or formula). This mathematical function is for the amount of money that Jean would spend at each of the stores. To do this, let’s think about the patterns in total cost for each of the two stores. To help us observe those patterns, we can make a table of costs. In this table, define x and y as the following variables:

• x = number of pounds of meat purchased • y = cost of purchasing the meat at Fresh-‐Plus.

The table below allows x to vary from 1 to 5 pounds. 1 Complete the next 2 calculations in the table. Fill in the value in the Fresh-‐Plus Cost column.

x y (computation) Fresh-‐Plus Cost x = 1 y = 0.5 + 4.50 $5.00 x = 2 y = 0.5 + 2·4.50 $9.50 x = 3 y = 0.5 + 3·4.50 $14.00 x = 4 x = 5

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2 Look at the pattern in the formula. How does the cost change each time Jean buys one more pound?

3 Imagine that x pounds of meat are purchased from Fresh-‐Plus.

If we don’t know the value of x, we can still represent the cost (y) with a formula – a mathematical function. Look at the y computations in the table. Use these computations to determine the formula for the cost of x pounds of meat.

y =

Let x continue to be pounds of meat purchased, but now let’s define y = cost for the warehouse store. 4 Make a table like the one above. Write an equation for y = cost of buying x pounds of meat from

the warehouse store.

x y (computation) Warehouse Cost

x = 1 x = 2 x = 3 x = 4 x = 5

Formula: y =

5 Write the equations for the Fresh-‐Plus and warehouse store costs. Then, graph them together

below. Fresh-‐Plus Cost Warehouse Cost

y = ________________ y = ________________




6 Do the graphs cross? If not, extend the lines until they do. 7 What is important about the point where the graphs cross? Describe the costs before the lines

cross and after the lines cross.

8 Under what circumstances would it would make financial sense to make the trip to the warehouse store to buy meat?

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Finding a Linear Function from Two Points When the graph of a mathematical function is a line, we call it a linear function. People put money in different types of investments, like savings accounts, government bonds, and the stock market. These types of investments have different types of interest, depending on the investment terms. One type is called simple interest. Simple interest pays you interest on your principle alone. (Note: you do not need to know all the details about interest right now, just remember that with simple interest the rate will not change.)

Suppose you invest a certain amount of money in an account that earns simple interest. At year 2 you have $1200. At year 6, you have $1400. We will find the linear function for account balance after x years.

We can find a linear function in four steps:

i Decide how to label the explanatory variable and the response variable, and choose units for them.

ii Find the slope. The slope is the rate of change of the response variable per unit change in the explanatory variable. This means how much y changes as x changes by 1 unit. We use the letter m for slope.

iii Find the y-‐intercept of the response variable. This is the value when the explanatory variable is zero. The y-‐intercept is the also called the initial value. We use the letter b for the y-‐intercept.

iv Write the formula for the line, in the form y = mx + b, using the values you found for m and b in steps ii and iii above. This formula is our mathematical model.

Below, we apply these steps to our simple interest problem.

i We want a function for computing account balance at any time, so let’s define y = account balance, and x = time. In context, it makes sense to measure y in dollars and x in years.

ii We can find the slope by computing the ratio of changes in each variable.

! =!ℎ!"#$ !" !!ℎ!"#$ !" !

=1400 − 1200 !"##$%&

6 − 2 !"

=200 !"##$%&

4 !"

= 50 !"##$%&/!"

= $50 !"# !"#$




iii Find the initial value, also known as the y-‐intercept.

We know that m is the slope and b is the y-‐intercept. Because this is a function, any point on the graph must satisfy this equation. This means we can substitute the coordinates of any point on the line for x and y in the equation and the equation will be true. We can solve for the y-‐intercept, b, if we make such a substitution.

Two pieces of information are given in the problem. We are told that when x = 2 years, the account balance is y = 1200.

! = !" + !

1200 = 50 ⋅ 2 + !

1200 = 100 + !

Notice the 100 + b on the right side. If we subtract 100 from each side, the 100s on the right will zero-‐out.

We will have solved for b.

1200 − !"" = 100 + ! − !""

1100 = !

iv We now have the slope, m, is 50 and the y-‐intercept, b, is 1100. Therefore, the formula for our linear function is y = mx + b.

y = 50x + 1100.

9 What is the initial value of the investment?

10 The equation found above can give two pieces of information:

• data values of the explanatory variable which represent times and

• data values of the response variable which represent the corresponding account balances.

Is the equation a statistical model of these points, or are they part of a mathematical function?




11 Explain what the slope tells us about the value of the investment over time.

YOU NEED TO KNOW

Be careful to subtract values in the right order when computing slope. The slope formula helps with this.

If the points, (x1, y1) and (x2, y2) are on a line, the slope of the line is

! =!ℎ!"#$ !" !!ℎ!"#$ !" !

=!! − !!!! − !!

Once we have seen the units, we don’t need to include them at each step as we did with the dollars and years in the last problem.

The units of slope are always units of y per unit of x. For example, the units in question 11 were dollars per year.

TRY THESE 12 Gary borrowed money from his father. He is paying him the same amount each month until the loan is

paid off. After 3 months, Gary owes $224. After a total of 7 months, Gary owes $160. Because the loan is being paid at a constant rate, the amount owed (y) is a linear function that depends on the number of months (x).

A Find the slope of the linear function. The linear function here gives the amount owed, y, after

x months.




B The equation of a linear function is y = mx + b. You know the slope, m, for this example. Use the slope and one of the points to find the value of b.

C Give the linear function which gives the amount owed after x months. D What is the amount owed at 1 year? Use the correct units in your answer.

We can find the equation of a linear function by looking at its graph. We just need to identify two points. Try the following problems.

TRY THESE

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13 Choose two points on the line. Then use the steps above to find the equation of the line. 14 Use the equation to predict y when x = 8.5. 15 Use the graph to estimate x if y = 6. Then use the formula to check whether your estimate is

correct.




TAKE IT HOME 1 If there is no wind resistance, a falling object’s speed increases at a constant rate.

Suppose you throw a watermelon down from the top of the Leaning Tower of Pisa in Italy. After 1 second it is traveling at 42 feet per second, and after 3 seconds it is travelling at 106 feet per second. A Find the find the equation of the line which gives the speed of the watermelon, y, after some

time x seconds. B How fast is the watermelon falling after 2.75 seconds? Use the proper units in your answer. C Does your equation represent a mathematical function or a statistical model?

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D Graph your equation below.

E What is the initial or beginning speed of the watermelon? Use the proper units in your answer.

2 One important model in economics is “supply and demand.” One principle of supply and demand

tells us that when a product’s price increases, demand goes down. For example, imagine the price of a product, like cell phones or shoes, goes up. People are less likely to want to pay more for the product that used to be cheaper. Their demand for the product will go down.

For this example, we will use the price for a bushel of corn. A bushel is the unit used for volume for dry products like corn or apples. It is equivalent to about 35 liters or 8 gallons. (Note: you do not need to convert the units in this problem. The equivalent units are given just to give you an idea of the size.) When the price of a bushel of corn increases, demand goes down. For the price of a bushel of corn, this rate of decrease is linear.

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4 million bushels of corn are demanded (x = 4) when the price per bushel is $2.15 (y = 2.15). 2 million bushels of corn are demanded (x = 2) when the price is $8.49 per bushel (y = 8.49). A Find the slope of the linear function. The linear function gives the price per bushel (y) that

corresponds to a demand of x (million) bushels of corn. B The equation of a linear function is y = mx + b. You know the slope, m, for this example. Find

the value of b, and C Write the linear function. The linear function gives the price that corresponds to a demand of

x bushels of corn. D What price corresponds to a demand of 3.5 million bushels? Use the proper units in your

answer. D Does your equation represent a mathematical function or a statistical model?




E Graph your equation below.

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3 Use the graph below to answer the following questions.

A Find the equation of the line. B Use the equation to find the y-‐intercept. C Use the graph to estimate x when y = 10. Then use the formula to check your estimate.

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This lesson is part of STATWAY®, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the Statway® Networked Improvement Community, please visit carnegiefoundation.org.

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lesson 12.1.2 version 2.5-student - De Anza Collegenebula2.deanza.edu/~mo/Math217/04-lesson_12.1.2... · statway®"studenthandout"""|"3""" lesson"12.1.2"" linearfunctions!! ©2013!thecarnegie!foundation!forthe!advancement!of!teaching!