Linear Functions

Objectives

By the end of this lesson you will be able to:

• Recognize a linear function.
• Given the ideal situation for a linear function, find the formula of the function and use it to solve word problems.

Introduction

In the linear equation tutorial, we learned to recognize, represent and use linear relations. In this section, we will be looking at a couple of examples, but instead of linear relations, we will be looking at linear functions.

In the first example, Juan has a job and earns $10 per hour.

x = work hours 0 1 2 3 4 f(x) = dollars earned 0 10 20 30 40 When the input increases by 1, the output increases by 10. The rate of change is 10 and is constant. When the rate is constant, we have learned in linear equations that the rate can be called slope y and the relation is linear.

The table and graph demonstrate the function:

Let's look at another example. Anne opened an account with the bank, deposited $20, and deposits $10 every week.

x = weeks passed 0 1 2 3 4 f(x) = money in the account 20 30 40 50 60 We begin with $20 (when x = 0, f(x) = 20), therefore the intercept of y is 20. When x increases by 1, y increases by 10 so the slope is 10.

The table and graph demonstrate the function:

Definition

The function

is linear if when increasing the input by 1, then the output increases by a constant value.

• The constant by which y increases when x increases by one is the slope. It's usually represented by the letter m.
• The value of f(x) when x = 0 is called the y-intercept and is usually represented by the letter b.

Create Linear Functions

Example 1:

Ajay washes cars. For every car he washes, he earns 20 dollars. He has 40 dollars to start with.

1. Create a function that shows the amount of cars Ajay washes and the amount of money he earns.
2. Use the function to determine how much money he has after he washes 25 cars.

Solution:

The following table shows the values of the the function:

Here we see a linear function.

• When the value of the input is 0, the output value is 40 which means that the y-intercept is 40.
• When the value of the input increases by 1, the value of the output increases by 20, in other words, the slope is 20.

The previous table can be rewritten as f(x) equals 40 dollars plus the number of cars washed multiplied by 20 dollars.

From that table we are able to obtain:

f(x)= 40 + 20x

Now we can use the formula to determine how much money Ajay will have after washing 25 cars:

f(25)= 40 + 20(25) = 540

Example 2:

Acme Company buys a machine for $12000. The annual depreciation value of the machine is $2000.

1. Create a function in which the input is the number of years owned and the output is the actual value of the machine.
2. Use the function to determine when the value will reach $0.

Solution:

The following table shows certain values of the function:

We have a linear function.

• When the input is 0, the output is 12000, making the y-intercept 12000.
• When the input is increased by 1, the value decreases by 2000, in other words the slope is -2000.

The previous table expressing f(x) can be written in the following manner:

From that table we are able to obtain the formula of the function:

f(x)= 12000 - 2000x

Now we can use the formula to determine when the machine will be worth $0:

0 = 12000 - 2000(x)

Looking for x, we obtain x = 6. Therefore 6 years after purchase, the machine will have lost its value.

Example 3:

A school director analyzes student enrollment. The year the school was founded, the school started with 400 students. Since then, 50 new students have been enrolling each year.

1. Create a function that has for input the number of years passed and for output the number of students.
2. Use the function to determine the number of students there will be after 15 years.

Solution:

The following table shows certain values of the function:

Here we have a linear function:

• When the input is 0, the output is 400. The intercept is 400.
• When the input is increased by 1, the output increases by 50, in other words, the slope is 50.

The previous table expressing f(x) may be rewritten in the following manner

From that table, it is possible to obtain the formula of the function:

f(x)= 400 + 50x

Now we are able to use the formula to determine how many students there will be after 15 years:

f(15) = 400 + 50(15) = 1150

The school will have 1150 students after 15 years.

Summary

Now that you have completed the section you are able to:

• Recognize a linear function.
• Given an appropriate situation for a linear function, find the formula of the function and use it to solve problems.