Interpret a Linear Model

Understand the idea

A linear model has the form y = mx + b. Slope m describes the change in y for each one-unit increase in x; b is the value when x is zero. Units explain what those values mean. A model applies within assumptions and a sensible domain. An intercept need not describe a realistic situation if zero is outside the context.

Worked example

A fictional service costs $12 plus $3 per hour: C = 3h + 12. The slope is $3 per hour and the intercept is the $12 starting fee. At h = 5, C = 3(5) + 12 = $27. Because the initial fee is not zero, cost is linear but not directly proportional to hours. The model assumes the same hourly price throughout the relevant period.

Try it yourself

1. Find the cost for 8 hours.
2. Solve for the hours if the cost is $42.
3. A second service costs $5 per hour with no fee. At what number of hours do costs match?

Check your reasoning

1. C = 3(8) + 12 = $36.
2. 42 = 3h + 12, so h = 10 hours.
3. 3h + 12 = 5h; 12 = 2h; h = 6 hours. Both cost $30.

Make it work for your learner

For support, model the example one step at a time and ask the learner to explain each choice in their own words. Use a sketch, organizer or physical model where helpful. For a challenge, create a new example, test an alternative explanation or describe when the model would not apply.

Exit check

Interpret slope and intercept and evaluate whether a linear model fits a context. Explain your reasoning with a specific example before moving on.

Interpret average change

For the model C(h) = 12 + 3h, a table gives C(0) = 12, C(2) = 18 and C(4) = 24. Between 0 and 4 hours, the average rate of change is (24 − 12) ÷ (4 − 0) = 3 dollars per hour. The starting fee is $12. Calculate the average rate between 2 and 4 hours and explain why it is the same.

Check: (24 − 18) ÷ (4 − 2) = 3 dollars per hour; this linear model has a constant rate.