Slope-Intercept Form (y = mx + b)
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what slope-intercept form is, how to graph lines from y = mx + b, and how to interpret slope and y-intercept in real-world situations.
The Concept
Section titled “The Concept”Slope-intercept form is the equation y = mx + b, where:
- m = slope (rate of change: how much y changes per unit of x)
- b = y-intercept (the value of y when x = 0; where the line crosses the y-axis)
This form makes graphing easy: start at y-intercept b, then use slope m to move up/down and right/left.
Steps to graph y = mx + b:
- Plot the y-intercept point (0, b)
- From that point, use slope m = rise/run
- Positive m: up and right
- Negative m: down and right
- Fraction m = p/q: up p, right q (or down if negative)
- Plot a second point, then draw a straight line through both
Example: y = 2x + 3
- y-intercept b = 3 → point (0, 3)
- Slope m = 2 = 2/1 → from (0, 3): up 2, right 1 → point (1, 5)
- Draw line through (0, 3) and (1, 5)
Worked Example
Section titled “Worked Example”Graph y = −3x + 6.
- y-intercept b = 6 → point (0, 6)
- Slope m = −3 = −3/1 → from (0, 6): down 3, right 1 → point (1, 3)
- Draw line through (0, 6) and (1, 3) (continues down-right)
Another: y = (1/2)x − 4
- y-intercept b = −4 → point (0, −4)
- Slope m = 1/2 → from (0, −4): up 1, right 2 → point (2, −3)
- Draw line through (0, −4) and (2, −3)
Real-World Application
Section titled “Real-World Application”Slope-intercept form models linear relationships:
- Pay: y = 18x + 50 (18 dollars per hour + 50-dollar bonus; y = total pay, x = hours)
- Cost: y = 3.50x + 20 (3.50 dollars per gallon gas + 20-dollar delivery fee)
- Savings: y = 0.05x + 200 (5% interest per year + 200 dollars initial; y = amount, x = years)
- Distance: y = 65x (65 mph driving; y = miles, x = hours)
Slope = rate of change (pay rate, cost per unit, interest rate)
y-intercept = starting amount (bonus, fee, initial deposit)
This form helps predict totals, compare plans, or plan budgets.