Finding Equations from Points, Tables, and Graphs

3.3: Finding Equations from Points, Tables, and Graphs. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 3 of 63 parts Practice Flashcards

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Line through two points

Objective: Determine an intercept after calculating slope.

Prerequisite skills: slope formulas and substitution. Two distinct points with different x-coordinates determine one nonvertical line. First compute the slope, then substitute either point into y = mx + b to find b. Common error: using y₁ directly as the intercept when x₁ is not zero. Restrictions and interpretation: Use the same x and y meanings across a table, equation, and graph.

Formulas

$b = y₁ − mx₁$

Worked examples

Try itA line passes through (2, 69) and (5, 135). What is its y-intercept?Show answer

Slope = (135 − 69)/(5 − 2) = 22. Use the first point: b = 69 − 22(2) = 25. Therefore the requested answer is 25.

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Line through two points

Objective: Determine an intercept after calculating slope.

Formulas

$b = y₁ − mx₁$

Worked examples

Try itA line passes through (2, 69) and (5, 135). What is its y-intercept?Show answer

Slope = (135 − 69)/(5 − 2) = 22. Use the first point: b = 69 − 22(2) = 25. Therefore the requested answer is 25.

Linear tables

Objective: Infer a constant rate from equal or unequal input intervals.

A table is linear when every output change divided by its corresponding nonzero input change is the same. Equal output differences alone establish linearity only when the input steps are equal. Common error: assuming unequal input steps must have equal output differences. Restrictions and interpretation: Use the same x and y meanings across a table, equation, and graph.

Formulas

(Δy)/(Δx) = constant for a linear table

Worked examples

Try itA linear table includes x-values 1, 3, 6 with corresponding y-values 31, 79, 151. What is y at x = 9?Show answer

From x = 1 to 3, the rate is 48/2 = 24. Thus y = 24x + 7. At x = 9, y = 223. Therefore the requested answer is 223.

Graph information without a picture

Objective: Recover a line from its intercepts.

The x-intercept has y = 0 and the y-intercept has x = 0. Treat these intercepts as two points to compute slope or an equation. A textual specification can replace a diagram when it gives all required information. Common error: using the two intercept numbers as if they were one point. Restrictions and interpretation: Use the same x and y meanings across a table, equation, and graph. 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

$intercepts (a,0) and (0,b) give m = −b/a, a ≠ 0$

Worked examples

Try itA nonvertical line crosses the axes at (25, 0) and (0, 75). What is y on the line when x = 1?Show answer

Slope = (0 − 75)/(25 − 0) = −3. The y-intercept is 75, so y = −3x + 75. At x = 1, y = 72. Therefore the requested answer is 72.

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