SAT Math Linear Functions

Slope, intercepts, line equations, and parallel/perpendicular relationships

Linear functions are the foundation of SAT Math. Every test has 4–6 questions directly about them, plus dozens more where they show up as a tool. If you're solid on lines, you have a strong base for systems, coordinate geometry, and any word problem with a constant rate of change.

The SAT tests linear functions in three forms: as a formula ( $y = m x + b$ ), as a graph, and as a word context ("flat fee plus per-minute charge"). All three reduce to the same task: find the slope $m$ and the y-intercept $b$ .

What the SAT actually tests

Computing slope from two points: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Writing line equations in $y = m x + b$ form
Finding x- and y-intercepts
Interpreting $m$ and $b$ in word context (rate of change / starting value)
Parallel lines (same $m$ ) and perpendicular lines (product of slopes $= - 1$ )
Point-slope form: $y - y_{0} = m (x - x_{0})$

Key concepts

Slope $m$

Steepness of the line. Numerically: how much $y$ changes when $x$ increases by 1. Positive — line rises; negative — falls; zero — horizontal.

y-intercept

The value of $b$ in $y = m x + b$ . The $y$ -value when $x = 0$ . In word problems, it's the "starting value."

Point-slope form

$y - y_{0} = m (x - x_{0})$ is faster when you have a point $(x_{0}, y_{0})$ and slope. The SAT favors this form — use it when a specific point is given.

Perpendicular lines

The product of slopes is $- 1$ . If one line has $m = 2$ , the perpendicular has $m = - \frac{1}{2}$ — flip and negate.

Worked examples

Example 1

Line $ℓ$ passes through $(2, 5)$ and $(6, 13)$ . What is the slope of $ℓ$ ?

Solution

Apply the slope formula: $m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$ .

💡 Point order doesn't matter as long as it's consistent in numerator and denominator.

Example 2

A line passes through $(3, 4)$ and is perpendicular to $y = 2 x + 1$ . What is its equation?

Solution

Given slope is $2$ , perpendicular slope is $- \frac{1}{2}$ . Use point-slope: $y - 4 = - \frac{1}{2} (x - 3)$ . Expanding: $y = - \frac{1}{2} x + \frac{3}{2} + 4 = - \frac{1}{2} x + \frac{11}{2}$ .

💡 Point-slope form is faster than computing $b$ from $y = m x + b$ . Memorize it.

Common pitfalls

Sign flips on slope — swapping numerator and denominator in $\frac{Δ y}{Δ x}$ .
Confusing $y = m x + b$ with standard form $a x + b y = c$ . Pick whichever form the question gives.
Computing perpendicular slope as $- m$ instead of $- \frac{1}{m}$ . Flip AND negate.
Dropping the negative sign on $- \frac{1}{m}$ . The SAT will catch you twice with the same mistake if you rush.

Exam strategy

Identify the form the question gives and mentally translate it to $y = m x + b$ . Two points? Use the slope formula. Point and slope? Point-slope form. Don't try to compute $b$ in your head when you already have a concrete point — point-slope saves about 20 seconds and prevents arithmetic errors. For word problems, read carefully for what the slope means: "cost goes up $5 per hour" means $m = 5$ .

Frequently asked questions

What is slope on the SAT?

Slope is rise over run: m = (y₂ − y₁)/(x₂ − x₁). Positive m — line rises; negative — falls; zero — horizontal.

How do I find a line equation from two points?

Compute the slope first, then use point-slope form: y − y₀ = m(x − x₀) with either point. Expand to y = mx + b if the question wants that form.

What's the slope of a perpendicular line?

The product of slopes is −1. If one line has m = 2, the perpendicular has m = −1/2. Rule: flip the fraction and negate.

How is y = mx + b different from ax + by = c?

Slope-intercept form directly shows slope and y-intercept. Standard form is easier for finding x-intercepts and solving systems. The SAT uses both — convert as needed.

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