Coordinate Plane, Slope, and Rate of Change

3.1: Coordinate Plane, Slope, and Rate of Change. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 63 parts Practice Flashcards

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Slope from two points

Objective: Compute slope with consistent coordinate order.

Prerequisite skills: coordinate pairs and subtraction. Slope measures vertical change per unit horizontal change. Subtract coordinates in the same order in numerator and denominator; reversing both orders preserves the ratio, but reversing only one changes its sign. Common error: reversing the subtraction order on only one coordinate. Restrictions and interpretation: Slope is undefined for a vertical line; use change in y divided by change in x.

Formulas

$m = (y₂ − y₁)/(x₂ − x₁), x₂ ≠ x₁$

Worked examples

Try itWhat is the slope through (2, 5) and (6, 93)?Show answer

The rise is 93 − 5 = 88. The run is 6 − 2 = 4. Divide: m = 88/4 = 22. Therefore the requested answer is 22.

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Slope from two points

Objective: Compute slope with consistent coordinate order.

Formulas

$m = (y₂ − y₁)/(x₂ − x₁), x₂ ≠ x₁$

Worked examples

Try itWhat is the slope through (2, 5) and (6, 93)?Show answer

The rise is 93 − 5 = 88. The run is 6 − 2 = 4. Divide: m = 88/4 = 22. Therefore the requested answer is 22.

Rate in context

Objective: Interpret slope with units.

For a linear model, slope is the change in output associated with one additional input unit. Its units are output units divided by input units. A negative slope describes a decrease. Common error: reporting the total change instead of the per-unit rate. Restrictions and interpretation: Slope is undefined for a vertical line; use change in y divided by change in x.

Formulas

rate = output change/input change

Worked examples

Try itA tank holds 140 liters at minute 2 and 265 liters at minute 7. It fills at a constant rate. What is the rate in liters per minute?Show answer

The volume rises by 125 liters over 7 − 2 = 5 minutes. The rate is 125/5 = 25 liters per minute. Therefore the requested answer is 25.

Recover a missing coordinate

Objective: Use a known slope to constrain an unknown point.

A known slope fixes the ratio between coordinate changes. Set the slope formula equal to the specified slope and solve the resulting linear equation. Check that the horizontal difference is nonzero. Common error: using the new x-coordinate rather than the change in x. Restrictions and interpretation: Slope is undefined for a vertical line; use change in y divided by change in x. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$y₂ = y₁ + m(x₂ − x₁)$

Worked examples

Try itA line of slope 25 passes through (3, 4) and (7, k). What is k?Show answer

The run is 7 − 3 = 4. Thus (k − 4)/4 = 25. Multiply by 4 and add 4: k = 104. Therefore the requested answer is 104.

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