Math Core

A taxi fare, a draining battery, a growing plant: many real situations change at a steady rate. A linear model is an equation $y = mx + b$ that describes one of these situations, so you can make predictions and answer questions about it.

Building a model from words

Every linear model needs a rate of change and an initial value. Look for them in the words.

Building a linear model

Decide what $x$ (the input) and $y$ (the output) stand for, with units.
Find the rate of change $m$ : the amount $y$ changes per 1 unit of $x$ .
Find the initial value $b$ : the value of $y$ when $x = 0$ .
Write $y = mx + b$ , and check that it makes sense.

Worked example: A taxi ride

A taxi charges $3 to start the ride plus $2 per mile. Write a model for the cost, and find the cost of an 8-mile ride.

Let $x$ be the miles and $y$ the cost in dollars. The rate is $2 per mile, and the initial value is $3 (the cost before you go anywhere).

y = 2x + 3

For 8 miles: $y = 2(8) + 3 = 19$ . The ride costs $19.

Taxi cost in dollars for x miles.Open in grapher →

Building a model from two data points

Sometimes you aren't told the rate or the starting value. You're given two measurements instead. Use them like two points on a line.

Worked example: A burning candle

A candle burns at a steady rate. After 2 hours it is 16 cm tall. After 5 hours it is 10 cm tall. Write a model for its height, and find when it burns out.

Let $x$ be hours and $y$ be height in cm. The points are $(2, 16)$ and $(5, 10)$ .

Rate: $m = \dfrac{10 - 16}{5 - 2} = \dfrac{-6}{3} = -2$ . The candle loses 2 cm per hour.

Initial value: $16 = -2(2) + b$ , so $16 = -4 + b$ and $b = 20$ . The candle started 20 cm tall.

y = -2x + 20

Burns out when $y = 0$ : $-2x + 20 = 0$ , so $x = 10$ . The candle lasts 10 hours.

Candle height in cm after x hours. It burns out at 10 hours.Open in grapher →

Common mistake

A negative rate of change means the quantity is decreasing. Keep the negative sign in your equation. Writing $y = 2x + 20$ would mean the candle grows taller as it burns.

Interpreting a model

Each part of a model has a real meaning. Always describe it with units.

Worked example: What do the numbers mean?

A plumber's bill is $C = 65h + 50$ , where $h$ is hours worked and $C$ is the cost in dollars.

65 is the rate of change: the plumber charges $65 for each hour.
50 is the initial value: a $50 charge even for 0 hours, such as a trip fee.
A 3-hour job costs $65(3) + 50 = 245$ dollars.

Describing a graph in words

You can read a story from a graph even without an equation. Look at where it goes up, goes down, or stays flat.

Maya's distance from home (km) over time (hours).Open in grapher →

Worked example: Reading a trip

The graph shows Maya's distance from home during a bike ride. Describe it.

From 0 to 3 hours, her distance increases steadily: she rides away from home at $\dfrac{6}{3} = 2$ km per hour.
From 3 to 5 hours, the graph is flat: she is 6 km from home and not moving. Maybe she stops for lunch.
From 5 to 8 hours, her distance decreases to 0: she rides home, again at 2 km per hour.

Tip

On a distance-time graph, a steeper piece means faster motion, and a flat piece means no motion at all. A flat piece does not mean the person is back home.

Practice

Practice 1

A gym charges a $25 sign-up fee plus $30 per month. Write a model for the total cost $y$ after $x$ months.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

Using the gym from the last problem, how many dollars does it cost for 6 months?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A plant grows at a steady rate. After 3 weeks it is 11 cm tall. After 7 weeks it is 19 cm tall. How tall was it at week 0, in centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Write a model for the plant's height $y$ (in cm) after $x$ weeks, using your answers to the last problem.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

A phone starts at 100% battery and loses 8% each hour. The model is $y = 100 - 8x$ . After how many hours is the battery at 20%?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A shop's cost to print $n$ custom T-shirts is $C = 4n + 15$ dollars. What does the 15 mean?

Practice 7

The graph shows a car's speed during a short trip. Which description matches?

Speed (mph) over time (minutes).Open in grapher →

Practice 8

A water tank drains at a steady rate. After 4 minutes it holds 340 gallons. After 10 minutes it holds 220 gallons. After how many minutes (from the start) is the tank empty?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.