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PREPITUS
AlgebraFoundational

Linear Functions

Learn how the SAT connects equations, graphs, tables, slope, and intercepts — and how to move between all four without redoing the algebra.

⏱ 8 min📍 Learn first: Linear Equations in Two Variables
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Free KC Learning Toolkit

Learn Linear Functions in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:40

The Linear Math of Taxi Meters

A short spoken refresher on this concept.

00:00 / 23:04

Linear Functions from Taxis to DSAT

The full lesson, worked through in depth.

Concept Snapshot

Definition
A linear function has a constant rate of change — its graph is a straight line, and it can always be written as f(x) = mx + b.
What the SAT tests
Translating between an equation, a graph, a table, and a real-world description of the same relationship.
Key rule
f(x) = mx + b · m = slope = rise/run · b = y-intercept = value at x = 0.
Most common mistake
Reading the y-intercept off a table that doesn't include x = 0, or swapping rise and run.
Prerequisite
Linear Equations in Two Variables

Visual Explanation

runriseb (y-intercept)xy
The straight line is the whole story: a constant rate of change (rise / run) and a starting value (b, where x = 0). Every equation, table, and word problem maps onto these two numbers.

Concept Mind Map

DefinitionA linear function has a constant rate of change — its graph is a straight line, and it can always be written as f(x) = mx + b.
Key rulef(x) = mx + b · m = slope = rise/run · b = y-intercept = value at x = 0.

Linear Functions

Algebra

Interpreting Slope in ContextAsks what a coefficient means in real-world units, not just its numeric value.
Table → EquationGives paired values and expects you to find m and b without a graph.
Graph → Equation MatchFour equations, one graph — distractors flip the sign of slope or intercept.

How the SAT Tests It

Pattern

Interpreting Slope in Context

Asks what a coefficient means in real-world units, not just its numeric value.

Clue: "per", "each", "for every"
Pattern

Table → Equation

Gives paired values and expects you to find m and b without a graph.

Clue: Two or more (x, y) rows
Pattern

Graph → Equation Match

Four equations, one graph — distractors flip the sign of slope or intercept.

Clue: "which equation represents"

Worked Example

A gym charges a $40 signup fee plus $15 per month. Which equation gives total cost y after x months?

We need a rule turning months into total cost — a function, y in terms of x.
Cost grows by the same amount every month → constant rate of change → linear function.
$15/month is the rate of change → m = 15. The $40 fee is paid before any months pass → b = 40.
y = 15x + 40. Check x = 0 → y = 40 ✓ matches the signup fee.

Try It Yourself

A ride-share fare starts at $3.50 and increases by $2.25 per mile. Which equation models the fare, y, for x miles?

Common Traps

What students do

Swapping m and b — putting the flat fee where the rate goes.

Why it fails

It changes what grows with x and what stays fixed — the model no longer matches the story.

Do this instead

Ask: what's true at x = 0? That value is always b. The number attached to "per" is always m.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Linear Functions from Taxis to DSAT

Slope, intercept, and reading tables — the full translation kit.

00:00 / 23:04

23:04 · the full lesson

What to notice

f(x) = mx + b · m = slope = rise/run · b = y-intercept = value at x = 0.

What the lesson covers

  • 0:00 Setup
  • 1:20 Slope
  • 2:45 Intercept
  • 4:00 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Linear Functions notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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