Topics
4. Functions & Linear Relationships
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United States of AmericaFL
Grade 912

4.05 Linear relationships

Lesson
Practice
Lesson

Concept summary

A function that has a constant rate of change is called a linear function. Linear functions can be written in the form:

\displaystyle f\left(x\right) = mx + b
\bm{m}
the slope of the line
\bm{b}
the y-value of the y-intercept

The graph of a linear function is a straight line.

In general, any relationship which has a constant rate of change is a linear relationship. A relationship which does not have a constant rate of change is called a nonlinear relationship.

-4
-3
-2
-1
1
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x
-4
-3
-2
-1
1
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y
A linear relationship
-4
-3
-2
-1
1
2
3
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x
-4
-3
-2
-1
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y
A nonlinear relationship

Worked examples

Example 1

Emanuel is selling raffle tickets to raise money for charity. The table below shows the cumulative number of tickets he has sold each hour for the first three hours:

Time (hours)123
Total ticket sales142842
a

State whether Emanuel's ticket sales represent a linear or nonlinear function.

Approach

A linear function will have a constant rate of change. We can compare the values in the table and see how much the total ticket sales are increasing by each hour.

Solution

Emanuel sells 14 tickets in the first hour. He then sells 28 - 14 = 14 tickets in the second hour, and 42 - 28 = 14 tickets in the third hour.

So the rate of change is constant and therefore the ticket sales represent a linear function.

b

Determine the rule which relates Emanuel's ticket sales and time.

Solution

From part (a) we know that Emanuel is able to sell 14 additional raffle tickets each hour.

c

If Emanuel's ticket sales continue in this way, determine the total number of tickets he will have sold after 6 hours.

Solution

From part (b) we know that Emanuel is selling 14 tickets per hour. So after 6 hours, if the pattern stays the same, he will have sold 14 \cdot 6 = 84 raffle tickets.

Example 2

Tiles were stacked in a pattern as shown:

An image showing tiles stacked in a pattern with increasing stack height and corresponding number of tiles. Stack 1 has 1 tile. Stack 2 has 3 tiles. Stack 3 has 5 tiles. And stack 4 has 7 tiles

A table of values representing the relationship between the height of the stack and the number of tiles was partially completed.

Height of stack1234510100
Number of tiles13
a

Identify the pattern for a relationship represented between the height of a stack and the number of tiles.

Solution

The number of tiles from one stack to the next increases by 2. The number of tiles can also be determined by taking the height of the stack, multiplying by 2, then subtracting 1.

b

Complete the table of values representing the relationship between the height of the stack and the number of tiles.

Solution

We can use the pattern identified in part (a) to help us fill in the table.

Height of stack1234510100
Number of tiles1357919199
c

Identify if the relationship between the stack height and number of tiles is linear or not.

Solution

There is a constant rate of change in the table of values and the pattern described a constant increase, so the relationship is linear.

Outcomes

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

MA.912.F.1.8

Determine whether a linear, quadratic or exponential function best models a given real-world situation.

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