Math

Linear Functions

2 min readEasy5-question drill

Linear functions are everywhere on the Math section — they show up as function notation, word problems about cost and time, and graphs. Master them and you'll bank easy points fast.

Where this lesson fits

Linear Functions

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→Linear Equations in 2 Variables →Systems of 2 Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables

A linear function is a rule that turns an input into an output, where the relationship graphs as a straight line. The standard form is f(x) = mx + b.

Let's break down that notation:

f(x) is read "f of x" — it just means "the output when the input is x." The f is the name of the function; the x inside the parentheses is the input.
m is the slope — how much the output changes every time x increases by 1. It's the rate of change.
b is the y-intercept — the output when x = 0. On a graph, it's where the line crosses the vertical axis.

Evaluating a function means plugging a number in for x and computing the output. If f(x) = 4x - 3, then f(2) means "replace every x with 2":

f(2) = 4(2) - 3 = 8 - 3 = 5

That's it. The number inside the parentheses goes everywhere you see x.

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Graph of f(x) = 4x - 3. The line crosses the y-axis at -3 (the intercept) and rises 4 for every step right.

Slope tells you steepness and direction. Positive slope rises left-to-right; negative slope falls. You can compute it between two points (x1, y1) and (x2, y2) with:

m = (y2 - y1) / (x2 - x1)

That's "change in output over change in input" — often called rise over run.

Slope as rise over run: from (1,4) to (3,10), rise = 6, run = 2, so slope = 3.

In word problems, the slope is the per-unit rate (cost per item, miles per hour) and the y-intercept is the starting amount (a flat fee, an initial value). For example, if a gym charges $30 to join plus $15 per month, total cost is C(m) = 15m + 30, where m is months. The 15 is the monthly rate; the 30 is the one-time fee.

Key skills the test checks:

Evaluate — plug in a value for x.
Interpret — identify what the slope or intercept means in context.
Build — write f(x) = mx + b from a graph, a table, or a description.

Watch your signs carefully when b is negative, and remember the input always goes inside the parentheses while the output is what comes out.

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Quick check

Check your understanding with a question from this topic:

If f(x) = 4x + (-3), what is f(4)?

Worked examples

Example 1

If f(x) = 5x - 3, what is f(7)?

Example 2

A linear function g passes through the points (1, 4) and (3, 10). What is the slope of g?

Example 3

A delivery service charges a flat fee plus a rate per mile. The total cost in dollars is given by C(x) = 1.5x + 6, where x is the number of miles. What is the meaning of the number 6 in this function?

Common pitfalls

Mixing up which number is the input

In f(7), the 7 is the input that replaces x — not the output. Students sometimes solve for when f(x) = 7 instead. Read whether the value is inside the parentheses (input) or set equal to f(x) (output).

Dropping a negative sign

With f(x) = 4x - 3, forgetting the minus gives the wrong answer. Rewrite subtraction clearly and keep the sign attached to b through every step.

Flipping the slope formula

Slope is rise over run: (y2 - y1)/(x2 - x1), not (x2 - x1)/(y2 - y1). And subtract the coordinates in the same order top and bottom, or you'll get the wrong sign.

Confusing slope with intercept in word problems

The per-unit rate (per mile, per month) is the slope m; the one-time or starting amount is the intercept b. Ask 'does this change with x, or is it fixed?' to tell them apart.

Key takeaways

A linear function has the form f(x) = mx + b: m is slope (rate), b is the y-intercept (starting value).
To evaluate f(a), substitute a for every x and follow order of operations.
Slope between two points = (y2 - y1)/(x2 - x1) = rise over run.
In context, slope is the per-unit rate and the y-intercept is the flat/initial amount.
Watch negative signs and don't confuse the input (inside parentheses) with the output.

Watch & learn

Curated Khan Academy walkthroughs on Linear Functions. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Linear Functions, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Linear Functions

You are here

→Linear Equations in 2 Variables →Systems of 2 Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables