Linear Functions

SAT Math — Linear Functions ## Slope-Intercept Form The most important form of a linear equation is slope-intercept form: y = mx + b - m = slope (rate of change) - b = y-intercept (where the line crosses the y-axis, i.e., the value of y when x = 0) Reading the equation: > y = 4x − 7 > Slope = 4 (y increases by 4 for every 1-unit increase in x) > Y-intercept = −7 (the line crosses the y-axis at (0, −7)) ## Calculating Slope Slope = rise / run = (y₂ − y₁) / (x₂ − x₁) Given two points (1, 3) and (4, 9): > m = (9 − 3) / (4 − 1) = 6/3 = 2 Slope tells you: - Positive slope: Line goes up left to right - Negative slope: Line goes down left to right - Zero slope: Horizontal line (y = constant) - Undefined slope: Vertical line (x = constant) — can't be expressed as a function ## Writing the Equation of a Line Given slope and y-intercept: Plug directly into y = mx + b. Given slope and one point: 1. Use slope-intercept form: y = mx + b 2. Plug in the known point (x, y) and slope m 3. Solve for b Example: Slope = 3, passes through (2, 8) > 8 = 3(2) + b → 8 = 6 + b → b = 2 > Equation: y = 3x + 2 Given two points: 1. Find the slope using the slope formula 2. Use slope and one point to find b (as above) ## Interpreting Linear Functions in Context The SAT loves to ask:…