Linear Functions in Context: Write, Interpret, Check | Grade 8 Math | LevelKite

Some stories follow a steady pattern. There is a starting amount, and then the same change happens again and again. A bowling night works this way when you pay once for shoes and then the same price for every game. A burning candle works this way when it loses the same height every hour.

In this lesson you turn stories like these into a rule, explain what each number in the rule means, and catch mistakes in a rule that someone else wrote.

Three terms first:

A function is a rule that turns each permitted input (a value you put in) into exactly one output (the value that comes out).
The domain is the set of permitted inputs.
A linear function is a function in which equal increases in the input always produce equal changes in the output. Its points lie along a straight line.

Key concept

Two numbers describe a linear function

Every linear function can be written as $y = mx + b$ , where $x$ is the input and $y$ is the output.

$m$ is the rate of change: how much the output changes each time the input goes up by 1. Its unit is output units per input unit.
$b$ is the initial value: the output when the input is 0. It describes the story's starting amount whenever 0 is a permitted input.

Find these two numbers in a story, and you can write its rule.

y = mx + b

To build the rule for a story, answer four questions.

What are the two quantities? Name each one with its unit. The quantity that depends on the other is the output, and the other is the input. For example, a bowling cost depends on the number of games, so games are the input and cost is the output.
What is the rate of change? Find how much the output changes for each 1 unit of input.
What is the initial value? Find the output when the input is 0. It may be a one-time fee or a starting amount.
Which inputs make sense? They form the domain. A count of games must be a whole number, and a height cannot be negative.

Then test your rule. Every fact in the story must fit it.

Worked example

A fee plus a price per game

On discount night, a bowling alley charges 3 dollars once to rent shoes and 2 dollars for each game. You rent shoes and may bowl at most 4 games. Write a function for your total cost, and state its domain.

Name the two quantities. Your total cost depends on how many games you bowl. So the number of games is the input, called g, and the total cost in dollars is the output, called C.
$g = \text{number of games}, \quad C = \text{total cost in dollars}$
Find the rate of change. Every added game adds the same 2 dollars to your cost, so the cost changes by 2 dollars per game. This steady change makes the function linear.
$m = 2 \text{ dollars per game}$
Find the initial value. With 0 games you have paid only for the shoes, so the output at input 0 is 3 dollars.
$b = 3 \text{ dollars}$
Write the rule. Start from the initial value and add the rate of change once for every game. In the form y = mx + b, the input g takes the place of x and the output C takes the place of y.
$C = 2g + 3$
State the domain. A number of games is a whole number, and the alley allows at most 4. So exactly five inputs are permitted, and each one gives exactly one cost.
$g = 0,\ 1,\ 2,\ 3 \text{ or } 4$
Test the rule against the story. For 2 games you pay 3 dollars for shoes plus 2 dollars twice, which is 7 dollars. The rule gives the same amount.
$C = 2(2) + 3 = 4 + 3 = 7$

Answer

Your total cost in dollars for $g$ games is $C = 2g + 3$ , where $g$ can be $0$ , $1$ , $2$ , $3$ or $4$ .

The five points of

C = 2g + 3

, with games

g

across and total cost

C

in dollars up.

The figure above shows the whole bowling function. The horizontal axis counts the games, $g$ . The vertical axis shows your total cost, $C$ , in dollars. A point $(g, C)$ means that bowling $g$ games costs $C$ dollars in all.

Each point comes straight from the rule: $2(0) + 3 = 3$ , $2(1) + 3 = 5$ , $2(2) + 3 = 7$ , $2(3) + 3 = 9$ and $2(4) + 3 = 11$ . The point $(0, 3)$ sits on the vertical axis and shows the initial value. From each point to the next, you move 1 game to the right and 2 dollars up. That repeated step shows the rate of change.

There are exactly five points because the domain has exactly five inputs. The points are not joined, because the alley sells only whole games.

Some stories give two facts instead of the rate of change and the initial value. Each fact pairs an input with its output.

Two facts are enough only when the story also says the change is steady. A constant rate means the same change for every 1 unit of input, so the function is linear. Without that statement, two facts tell you nothing about other inputs.

Treat the two facts like two rows of a table, which you already know how to use:

The rate of change is the change in output divided by the change in input. The two inputs must be different: a function gives each input only one output, and you cannot divide by 0. (Two points with the same input and different outputs would lie on a vertical line, which is not the graph of a function.) Two different inputs may share an output. Then the rate of change is 0.
Once you know the rate $m$ , put one fact into $y = mx + b$ and solve for $b$ .

Worked example

Two facts about a candle

A candle burns down at a constant rate. It is 10 cm tall 1 hour after it is lit, and 4 cm tall 4 hours after it is lit. Write a function for the candle's height, and state its domain.

Name the quantities and list the facts. The height depends on the time. So the number of hours since the candle was lit is the input, called t, and the height in centimeters is the output, called h. Write each fact as a pair (t, h).
$(t,\ h) = (1,\ 10) \quad \text{and} \quad (t,\ h) = (4,\ 4)$
Find the rate of change. Divide the change in height by the change in time, and subtract in the same order on the top and on the bottom. The inputs 1 and 4 are different, so you are not dividing by 0.
$m = \frac{4 - 10}{4 - 1} = \frac{-6}{3} = -2$
Find the initial value. Put the rate and the fact (1, 10) into h = mt + b. Multiply first, then add 2 to both sides. Adding the same number to both sides keeps them equal.
$10 = -2(1) + b \quad\Rightarrow\quad 10 = -2 + b \quad\Rightarrow\quad 12 = b$
Write the rule, then test it with the fact you have not used yet. At 4 hours the rule must give 4 centimeters, and it does.
$h = -2t + 12 \qquad \text{test: } -2(4) + 12 = -8 + 12 = 4$
State the domain. Time starts at 0, when the candle is lit. A height cannot be negative, so the rule stops when the height reaches 0. To find that time, add 2t to both sides, then divide both sides by 2, which is not 0. Both moves keep the two sides equal.
$0 = -2t + 12 \quad\Rightarrow\quad 2t = 12 \quad\Rightarrow\quad t = 6$

Answer

The candle's height in centimeters is $h = -2t + 12$ , where $t$ is any number of hours from $0$ to $6$ . You can write this domain as $0 \le t \le 6$ .

A rule is more useful when you can say what its numbers mean. To interpret a number is to explain it in the words and units of the story.

For the rate of change, give the unit as output units per input unit and say which way the output moves. With a positive rate the output rises as the input grows. With a negative rate it falls. With a rate of 0 it stays the same.
For the initial value, give the output unit and say what input 0 means in the story.

Take the bowling rule $C = 2g + 3$ . The rate of change $2$ means your cost rises 2 dollars per game. The initial value $3$ means that with 0 games you have paid 3 dollars, the shoe rental.

Worked example

What the candle rule tells you

The candle's height is $h = -2t + 12$ for $0 \le t \le 6$ , where $t$ is the number of hours since the candle was lit and $h$ is its height in centimeters. Interpret the rate of change and the initial value.

Find the unit of the rate of change. The output is measured in centimeters and the input in hours, so the rate is measured in centimeters per hour.
$m = -2 \text{ cm per hour}$
Say what the rate means. Compare two times that are 1 hour apart, such as 2 hours and 3 hours. The height changes by -2 centimeters. The negative sign tells you the candle gets shorter.
$-2(2) + 12 = 8 \qquad -2(3) + 12 = 6 \qquad 6 - 8 = -2$
Find the initial value by using input 0, the moment the candle is lit. Time 0 is in the domain, so this output is a real height.
$h = -2(0) + 12 = 0 + 12 = 12$
Say what the initial value means. The candle was 12 centimeters tall at the moment it was lit. The story never stated this height. The rule revealed it.

Answer

The rate of change $-2$ means the candle gets $2$ cm shorter every hour, and the initial value $12$ means the candle was $12$ cm tall when it was lit.

Line A is

h = -2t + 12

, with hours

t

across and height

h

in centimeters up.

The figure above shows the candle rule. The horizontal axis shows $t$ , the hours since the candle was lit, and the vertical axis shows $h$ , its height in centimeters. A point $(t, h)$ means the candle is $h$ cm tall $t$ hours after it was lit.

Line A is the rule $h = -2t + 12$ . The story's two facts are the points $(1, 10)$ and $(4, 4)$ . Both fit the rule, since $-2(1) + 12 = 10$ and $-2(4) + 12 = 4$ . The initial value is the point $(0, 12)$ , where the line meets the vertical axis. The rate of change is the line's steady fall: from $(0, 12)$ to $(1, 10)$ you move 1 hour to the right and 2 cm down, and the same step repeats all along the line.

The function uses only the part of Line A from $(0, 12)$ to $(6, 0)$ , because the domain is $0 \le t \le 6$ and $-2(6) + 12 = 0$ . This time the points are joined, because every time in that range, such as $2.5$ hours, is a permitted input.

Worked example

Find and fix the error

A fish tank already holds some water. A hose then adds water at a constant rate for 10 minutes. The tank holds 9 liters 2 minutes after the hose is turned on, and 15 liters 5 minutes after it is turned on. Dev writes $V = 2t + 9$ , where $t$ is the number of minutes since the hose was turned on and $V$ is the number of liters in the tank. Find and correct his error.

Test Dev's rule with a fact from the story. At 2 minutes the tank holds 9 liters, but his rule gives 13 liters. The rule does not fit the story, so it contains an error.
$2(2) + 9 = 4 + 9 = 13 \ne 9$
Check his rate of change. The inputs 2 and 5 are different, so divide the change in liters by the change in minutes. The rate is 2 liters per minute, so the 2 in his rule is correct.
$m = \frac{15 - 9}{5 - 2} = \frac{6}{3} = 2$
Find the error. Dev used 9 as the initial value. It is the first amount in the story, but it is the amount at 2 minutes, not at 0 minutes. The initial value must be the output when the input is 0.
Correct the initial value. Put the rate and the fact (2, 9) into V = 2t + b. Multiply first, then subtract 4 from both sides. Subtracting the same number from both sides keeps them equal.
$9 = 2(2) + b \quad\Rightarrow\quad 9 = 4 + b \quad\Rightarrow\quad 5 = b$
Write the corrected rule and test it with both facts. It gives 9 liters at 2 minutes and 15 liters at 5 minutes, so it fits the story. The hose runs for 10 minutes, so the inputs go from 0 to 10.
$V = 2t + 5 \qquad 2(2) + 5 = 4 + 5 = 9 \qquad 2(5) + 5 = 10 + 5 = 15$

Answer

Dev used the amount at 2 minutes as the initial value. The correct function is $V = 2t + 5$ for $0 \le t \le 10$ , so the tank held 5 liters when the hose was turned on.

Line B is Dev's rule

V = 2t + 9

and Line C is the corrected rule

V = 2t + 5

, with minutes

t

across and liters

V

up.

The figure above shows Dev's error. The horizontal axis shows $t$ , the minutes since the hose was turned on, and the vertical axis shows $V$ , the liters in the tank. A point $(t, V)$ means the tank holds $V$ liters after $t$ minutes. The figure shows only the first part of the 10 minutes.

Line B is Dev's rule $V = 2t + 9$ , and Line C is the corrected rule $V = 2t + 5$ . The story's facts are the points $(2, 9)$ and $(5, 15)$ . Both fit Line C, since $2(2) + 5 = 9$ and $2(5) + 5 = 15$ . Neither fits Line B. Dev's rule gives the point $(2, 13)$ at $t = 2$ , and $2(5) + 9 = 19$ at $t = 5$ . Each is 4 liters too high.

The two lines rise at the same rate, because Dev's rate was right. Only the starting points differ. Dev's line starts at $(0, 9)$ , since $2(0) + 9 = 9$ , but 9 liters belongs at $t = 2$ . The corrected line starts at $(0, 5)$ , since $2(0) + 5 = 5$ .

Check by putting numbers into the rule, not by how the picture looks. When a story is linear, a rule $y = mx + b$ that fits two facts with different inputs is the correct rule, because only one straight line passes through two different points.

The takeaway: a linear story comes down to two numbers. The rate of change tells you how much the output changes for each 1 unit of input. The initial value tells you the output when the input is 0. Put them into $y = mx + b$ , explain each number in the story's own units, and state which inputs are permitted. Then test your rule against the story at two or more different inputs before you trust it.