2.1Linear Functions
Write linear functions in slope-intercept and point-slope form, interpret slope as rate of change, and identify parallel and perpendicular lines.
Linear functions are the simplest and most widely used models in mathematics. Every tangent line in calculus is linear, and linear approximation is one of the most powerful tools in all of applied math.
Essential Question
What makes a function linear, and how does the slope of a line describe the rate at which one quantity changes relative to another?
Lesson Overview
A linear function is any function of the form f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept (initial value). Its graph is always a straight line. The slope m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁) measures how steeply the line rises or falls: positive slope rises left to right, negative slope falls, zero slope is horizontal, and undefined slope is vertical. The slope-intercept form y = mx + b is best when you know the slope and y-intercept. The point-slope form y − y₁ = m(x − x₁) is best when you know the slope and one point. Parallel lines have equal slopes (m₁ = m₂) and different y-intercepts. Perpendicular lines have slopes that are negative reciprocals: m₁ · m₂ = −1. A linear function has constant rate of change — its slope is the same between any two points on the line.
Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)
- m > 0: line rises left to right
- m < 0: line falls left to right
- m = 0: horizontal line (y = b)
- m undefined: vertical line (x = h)
- |m| large: steep; |m| small: shallow
Writing Equations
Given slope m and y-intercept b:
y = mx + b
Given slope m and point (x₁, y₁):
y − y₁ = m(x − x₁)
Given two points (x₁,y₁) and (x₂,y₂):
Find m first, then use point-slope.
Slope-Intercept Form: y = 2x + 3
Forms of a Linear Equation
| Form | Equation | When to Use | Example |
|---|---|---|---|
| Slope-Intercept | y = mx + b | Know slope m and y-intercept b | y = 3x − 2 |
| Point-Slope | y − y₁ = m(x − x₁) | Know slope m and one point (x₁, y₁) | y − 4 = 2(x − 1) |
| Standard | Ax + By = C | Integer coefficients; intercepts easy to find | 3x − y = 2 |
| Horizontal | y = k | Slope = 0; parallel to x-axis | y = 5 |
| Vertical | x = h | Undefined slope; parallel to y-axis | x = −3 |
Parallel and Perpendicular Lines
Parallel Lines
Same slope, different y-intercepts. f ∥ g because both have m = 2.
Perpendicular Lines
Slopes are negative reciprocals: m₁ · m₂ = −1. f ⊥ h: 2 · (−½) = −1.
Worked Examples
Write the equation of the line with slope −3 and y-intercept 7 in slope-intercept form. Then find f(4).
Slope-intercept form: y = mx + b. Here m = −3 and b = 7.
Equation: y = −3x + 7.
Find f(4): f(4) = −3(4) + 7 = −12 + 7 = −5.
Write the equation of the line passing through (2, 5) with slope 4. Convert to slope-intercept form.
Use point-slope form: y − y₁ = m(x − x₁).
Substitute m = 4, (x₁, y₁) = (2, 5): y − 5 = 4(x − 2).
Distribute: y − 5 = 4x − 8.
Add 5: y = 4x − 3.
Find the equation of the line passing through (−1, 3) and (3, 11).
Find slope: m = (11 − 3)/(3 − (−1)) = 8/4 = 2.
Use point-slope with (−1, 3): y − 3 = 2(x − (−1)) = 2(x + 1).
Simplify: y − 3 = 2x + 2 → y = 2x + 5.
Check with (3, 11): y = 2(3) + 5 = 11 ✓.
Line ℓ has equation y = (1/3)x − 2. Write equations for (a) a line parallel to ℓ through (6, 1), and (b) a line perpendicular to ℓ through (0, 4).
Slope of ℓ: m = 1/3.
(a) Parallel: same slope m = 1/3. Through (6, 1): y − 1 = (1/3)(x − 6) → y = (1/3)x − 1.
(b) Perpendicular: slope = −1/(1/3) = −3. Through (0, 4): y = −3x + 4.
A taxi charges a $3.50 flat fee plus $2.25 per mile. Write a linear function C(m) for the total cost after m miles. Find the cost for 8 miles and interpret the slope.
Flat fee = $3.50 (y-intercept b). Rate = $2.25/mile (slope m).
C(m) = 2.25m + 3.50.
C(8) = 2.25(8) + 3.50 = 18.00 + 3.50 = $21.50.
Slope interpretation: the cost increases by $2.25 for each additional mile driven.
Guided Practice
Write the equation of the line with slope −2 and y-intercept 5. Find f(−3).
Hint: Use y = mx + b directly. Substitute m = −2, b = 5. Then plug in x = −3.
Write the equation of the line through (4, −1) with slope 3. Give your answer in slope-intercept form.
Hint: Start with point-slope: y − (−1) = 3(x − 4). Distribute and simplify to y = mx + b.
Find the slope of the line through (−2, 7) and (4, −5). Then write its equation.
Hint: m = (y₂ − y₁)/(x₂ − x₁). Then use point-slope with either point.
Are the lines y = 4x − 1 and y = −(1/4)x + 3 parallel, perpendicular, or neither?
Hint: Compare slopes. Parallel: equal slopes. Perpendicular: m₁ · m₂ = −1. Check: 4 · (−1/4) = ?
A plumber charges a $50 service fee plus $75 per hour. Write a linear function C(h) for total cost after h hours. How many hours were worked if the bill was $275?
Hint: C(h) = 75h + 50. Set C(h) = 275 and solve for h.
Key Vocabulary
Linear Function
A function of the form f(x) = mx + b. Its graph is a straight line. Has constant rate of change (slope) everywhere.
Slope (m)
Rate of change: m = Δy/Δx = (y₂−y₁)/(x₂−x₁). Measures steepness and direction. Positive = rising, negative = falling, zero = horizontal.
y-Intercept (b)
The point where the line crosses the y-axis (x = 0). In y = mx + b, b is the y-intercept. Represents the initial value.
Slope-Intercept Form
y = mx + b. Best when slope and y-intercept are known. Easiest form for graphing.
Point-Slope Form
y − y₁ = m(x − x₁). Best when slope and one point are known. Starting form when given two points.
Parallel / Perpendicular
Parallel lines: equal slopes (m₁ = m₂). Perpendicular lines: negative reciprocal slopes (m₁ · m₂ = −1).
Interactive Practice — 5 Questions
What is the slope of the line y = −5x + 9?
Which equation passes through (3, 7) with slope 2?
Find the slope of the line through (−1, 4) and (3, −4).
Which pair of lines is perpendicular?
A phone plan costs $25/month plus $0.10 per text. Which function models the monthly cost C for t texts?
Independent Practice
Independent Practice
Write the equation of the line with slope 6 and y-intercept −4. Find f(2) and f(−1).
Write the equation of the line through (−3, 5) with slope −2. Give your answer in slope-intercept form.
Find the equation of the line through (1, −2) and (5, 6).
Write equations for (a) a line parallel to y = −3x + 1 through (2, 4), and (b) a line perpendicular to y = −3x + 1 through (0, −1).
A car rental costs $40 per day plus $0.20 per mile. Write a linear function C(m) for total cost. Find the cost for 3 days and 150 miles.
Common Mistakes
Swapping rise and run: computing m = (x₂−x₁)/(y₂−y₁) instead of (y₂−y₁)/(x₂−x₁).
Slope = rise over run = Δy/Δx. Always put the y-difference in the numerator and the x-difference in the denominator.
Confusing parallel and perpendicular slopes: thinking perpendicular lines have the same slope.
Parallel lines have EQUAL slopes. Perpendicular lines have NEGATIVE RECIPROCAL slopes: flip and negate. Slope 3 → perpendicular slope is −1/3.
Using point-slope form incorrectly: writing y + y₁ = m(x + x₁) instead of y − y₁ = m(x − x₁).
Point-slope form subtracts: y − y₁ = m(x − x₁). If the point is (−2, 5), write y − 5 = m(x − (−2)) = m(x + 2).
Forgetting that a vertical line x = h has undefined slope, not zero slope.
Horizontal line y = k has slope 0. Vertical line x = h has UNDEFINED slope (division by zero in Δx = 0).
Math Tips
To find the y-intercept from slope-intercept form y = mx + b, just read off b. To find the x-intercept, set y = 0 and solve for x.
Perpendicular slope shortcut: flip the fraction and change the sign. Slope 2/3 → perpendicular slope −3/2. Slope −4 → perpendicular slope 1/4.
Two points always determine a unique line. Find slope first, then use either point in point-slope form.
Slope as rate of change: if C(t) = 5t + 20, the slope 5 means "cost increases by $5 per unit of t." Always attach units to your slope interpretation.
Standard form Ax + By = C is useful for finding intercepts quickly: set x = 0 for y-intercept, set y = 0 for x-intercept.