Slope and Linear Equations
Slope measures steepness: how much y changes for each unit change in x. It is the rate of change. Slope-intercept form (y = mx + b) is the most useful way to write a linear equation: m is the slope, b is where the line crosses the y-axis.
Key Concepts
Calculating Slope
Slope = rise/run = (y₂ − y₁)/(x₂ − x₁). From (1, 3) to (4, 9): slope = (9−3)/(4−1) = 6/3 = 2. Positive slope goes up left-to-right. Negative slope goes down. Zero slope is horizontal. Undefined slope is vertical.
Slope-Intercept Form
y = mx + b. m is the slope (rise/run). b is the y-intercept (where the line crosses the y-axis). From slope = 3 and y-intercept = −2: y = 3x − 2. Reading a graph: where does it cross y-axis? (b). How much does y change per x step? (m).
Parallel and Perpendicular Lines
Parallel lines have equal slopes (y = 2x+1 and y = 2x−3 are parallel). Perpendicular lines have slopes that are negative reciprocals: if m = 3, the perpendicular slope is −1/3. Their product is always −1.
🆕 Slope Explorer
Adjust slope and y-intercept. See how they change the line!
✅ Check Your Understanding
1. Slope is defined as...
2. In y = 3x − 5, what is the y-intercept?
3. Two lines with slopes 2 and −½ are...