Linear Functions and Slope
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what makes a function linear, how to find slope from two points or a graph, and what slope tells us about real-world rates of change.
The Concept
Section titled “The Concept”A linear function is a function whose graph is a straight line. Its equation is usually written as y = mx + b, where:
- m = slope (how steep the line is, rise over run)
- b = y-intercept (where the line crosses the y-axis when x = 0)
Slope (m) measures the rate of change: how much y changes for each unit change in x.
Slope formula from two points (x₁, y₁) and (x₂, y₂):
- Positive slope: line goes up left-to-right (increasing)
- Negative slope: line goes down left-to-right (decreasing)
- Zero slope: horizontal line (no change in y)
- Undefined slope: vertical line (no change in x)
Here’s what a line with slope 2 looks like. It passes through (2, 5) and (6, 13), rising 8 units over a run of 4:
Worked Example
Section titled “Worked Example”Find the slope between points (2, 5) and (6, 13).
- Rise = y₂ − y₁ = 13 − 5 = 8
- Run = x₂ − x₁ = 6 − 2 = 4
- m = 8 / 4 = 2
Slope = 2 (for every 1 unit right, y increases by 2 units).
Another: Points (4, 9) and (1, 15)
- Rise = 15 − 9 = 6
- Run = 1 − 4 = −3
- m = 6 / (−3) = −2
Slope = −2 (decreasing: for every 1 unit right, y decreases by 2 units).
Real-World Application
Section titled “Real-World Application”Slope represents rates in daily life:
- Pay: 18 dollars per hour → slope = 18 (y = pay, x = hours)
- Fuel efficiency: 28 miles per gallon → slope = 28 (y = miles, x = gallons)
- Cost: 3.50 dollars per gallon → slope = 3.50 (y = cost, x = gallons)
- Speed: 65 mph → slope = 65 (y = distance, x = time in hours)
Positive slope = increasing (earning more, growing savings)
Negative slope = decreasing (spending down balance, temperature dropping)
Understanding slope helps predict trends, compare options, and plan.