Linear Functions

Unit 1 · Algebra

What the SAT asks here

A linear function has a constant rate of change. The Digital SAT focuses on three things: computing slope from two points, interpreting m and b in a real-world context, and comparing slopes of parallel vs. perpendicular lines. Expect 2–3 problems per test.

Slope-intercept form

Standard form

y = mx + b

m = slope (rate of change), b = y-intercept (value when x = 0)

Slope from two points

Given (x₁, y₁) and (x₂, y₂)

m = \frac{y_2 - y_1}{x_2 - x_1}

Parallel vs. perpendicular

Parallel — same slope, different intercept

m_1 = m_2

Perpendicular — negative reciprocal slopes

m_1 \cdot m_2 = -1

SAT Tip: In a word problem the slope is the per-unit rate of change and the y-intercept is the starting amount at x = 0. Re-read the question to make sure you're returning the right one.

Caution: Perpendicular slope is the negative reciprocal, not just the reciprocal. If m = 2/3, perpendicular is −3/2.

Type 1

Slope from two points

Apply the slope formula directly: subtract y-coordinates over subtracted x-coordinates. Watch sign carefully — a minus-minus becomes plus.

Example 1

A line passes through (2, −1) and (6, 7). What is the slope of the line?

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Type 2

Interpret slope and intercept in context

Translate a real-world scenario into m and b. The slope is the per-unit rate; the y-intercept is the value at x = 0. Often the SAT gives you two data points and asks for the rate.

Example 2

A taxi charges a flat fee plus a per-mile rate. After 3 miles the fare is $12, and after 7 miles the fare is $20. What is the per-mile rate in dollars per mile?

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Type 3

Parallel and perpendicular lines

Parallel lines share the same slope. Perpendicular slopes multiply to −1, equivalently the negative reciprocal. Convert to slope-intercept first if you have to.

Example 3

Line ℓ has equation y = −(1/3)x + 4. What is the slope of a line perpendicular to ℓ?

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