Linear Function Applications
Model and solve real-world problems using linear functions: cost functions, depreciation, phone plan comparisons, and more. Write, graph, and interpret linear models from word descriptions and data tables.
A lesson of Algebra 1, in Linear Functions, about 30 minutes.
By the end of this lesson, you will be able to build a linear model from a situation, interpret its slope and intercept in the words of that situation, use it to answer questions in both directions, and say where the model stops being true.
The Model Is the Answer
Every applied linear question is the same three moves.
- Build it. Find the rate and the starting amount, and write $y = mx + b$
- Read it. Say what $m$ and $b$ mean, with units
- Use it. Given an input find the output, or given an output solve for the input
The third move goes both ways, and which way you are going decides whether the work is substitution or solving an equation. Noticing that before you start saves most of the confusion in this topic.
A gym charges $$40$ to join and $$25$ a month.
$$C = 25m + 40$$
"What does 6 months cost?" is substitution: $C = 190$.
"How many months for $$240$?" is an equation: $25m + 40 = 240$, so $m = 8$.
Same model, two directions, two different kinds of work.
Reading the Two Numbers Back
Marks live here, and the answer is always a sentence with units in it.
| Situation | $m$ means | $b$ means |
|---|---|---|
| Cost against months | dollars per month | the joining fee |
| Distance against hours | speed, in miles per hour | the head start |
| Water against minutes | liters per minute, filling or draining | what was there at the start |
| Value against years | depreciation per year, negative | the price when new |
The sign comes from the verb. "Loses $$1{,}500$ a year" is $m = -1500$; the sentence never writes the minus and the answer must.
$b$ is the output at zero, so it is the fee before any months, the water before any minutes, the value when new. If the situation has no sensible "zero", the intercept may not mean anything on its own, and saying so is a better answer than inventing a meaning.
Example 1: Building and Reading
Here is the whole method on one situation, worked all the way through.
A tank holds 300 liters and drains at 12 liters per minute.
- Write the rate as a signed number -12
- What is the starting amount? 300
- Write the model for liters L after t minutes L = -12t + 300