Linear Functions and Slope

A linear function produces a straight-line graph. The most useful form is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

Slope measures steepness and direction: m = (y₂ - y₁) / (x₂ - x₁). It tells you how much y changes for each 1-unit increase in x.

• Positive slope: Line rises left to right.
• Negative slope: Line falls left to right.
• Zero slope: Horizontal line (y = constant).
• Undefined slope: Vertical line (x = constant).

Point-slope form is handy when you know one point (x₁, y₁) and the slope: y - y₁ = m(x - x₁).

Parallel and perpendicular lines

• Parallel lines share the same slope: m₁ = m₂.
• Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = -1.

Real-world context: In y = 0.15x + 25, the slope 0.15 means cost increases $0.15 per mile, and the y-intercept 25 is the flat base fee.

⚠️ Common mistake: Mixing up x and y in the slope formula — always subtract y-values in the numerator and x-values in the denominator.

💡 Tip: Rewrite any linear equation in slope-intercept form first; the slope and intercept are then immediately visible.