Linear models show up constantly on the test, often dressed up as a real-world story about money, distance, or time. If you can read what the slope and the starting number actually mean, these become some of the fastest points on the whole section.
A linear model is just an equation with a straight-line relationship between two quantities. It usually looks like:
y = mx + b (or in function notation, f(x) = mx + b)
The test loves to wrap these in a real situation — cost, population, water in a tank — and then ask what a number means. Let's nail down every part.
x is the input (the thing you control or that changes): hours worked, number of months, miles driven.
y or f(x) is the output (the result): total pay, total cost, amount left.
m is the slope — the rate of change. It tells you how much y changes every time x goes up by 1. Units matter: dollars per hour, people per year. The word "per" is your slope signal.
b is the y-intercept — the starting value when x = 0. It's the flat fee, the initial amount, the value before anything happens.
Example: C = 25h + 40. Here C is total cost, h is hours. The 40 is a starting fee (cost before any hours, when h = 0). The 25 is the rate: cost goes up $25 per hour.
There are two main question styles:
Evaluate the model — they give you an x value, you plug it in. If f(x) = 4x - 3, then f(4) = 4(4) - 3 = 13. Just substitute and compute.
Interpret a coefficient — they point at the m or the b and ask what it represents in the story. To answer, attach units and ask:
Is it the slope (m)? Then it's a rate — "per" something.
Is it the intercept (b)? Then it's the starting amount when the input is 0.
The whole skill is translating between math symbols and plain English. Read the variable definitions the problem gives you, then map each number to its real-world meaning. No heavy algebra required — just careful reading and substitution.
If f(x) = 5x + (-3), what is f(7)?
A landscaping company charges customers using the model C = 35h + 50, where C is the total charge in dollars and h is the number of hours of work. What does the number 35 represent in this model?
A biologist models a deer population with P = 1200 - 45t, where P is the number of deer and t is the number of years since the study began. Which statement best interprets the model? (Choose the correct interpretation.)
Students grab the wrong number when asked what something 'represents.' The number attached to the variable (m) is the rate (per unit); the lonely constant (b) is the starting value when the input is 0. Identify which one they're pointing at first.
A model like P = 1200 - 45t has slope -45, meaning a decrease. If you read only the 45 you'll wrongly pick 'increases.' Always carry the sign.
In f(4) = 4(4) - 3, you must multiply before subtracting (16 - 3 = 13), not add then multiply. Do the multiplication first, every time.
The problem tells you what x and y stand for (hours, dollars, years). Interpretation answers must match those exact units — read the setup sentence before the math.
Linear models have the form y = mx + b: m is the rate of change (slope), b is the starting value (y-intercept).
The coefficient of the variable is a 'per unit' rate — find the word 'per' to describe it.
The constant term is the value when the input equals 0 (the initial/starting amount).
To evaluate a model, substitute the given input and follow order of operations (multiply before adding).
A negative slope means the quantity decreases as the input grows — always check the sign.