Interpreting slope & intercept

Concept

Whenever a real-world relationship is modelled by y = mx + b, the two numbers tell a story:

Slope m = rate of change. "For every 1-unit increase in x, y changes by m." Read it with units — dollars per hour, kilometres per minute, cm per week.
y-intercept b = starting value. The value of y when x = 0. In real-world terms: the fixed fee, the initial position, the upfront cost.

SAT word problems often ask "what does the 25 in this equation represent?" — your job is to translate the symbol back into a sentence with units.

Solution

Slope. The 25 sits next to h ⇒ rate of change = $25 per hour. Each extra hour adds $25.

y-int. The 40 stands alone ⇒ value when h = 0 = $40 fixed call-out fee, charged before any work begins.

Plug in. C = 25(3) + 40 = 75 + 40 = 115

A 3-hour job costs $115: $75 of labour plus the $40 call-out fee.

Solution

Rate. 10¢ per minute = $0.10 per minute ⇒ slope = 0.10.

Start. $30 base fee charged regardless of usage ⇒ y-intercept = 30.

Build. C = 0.10m + 30

Slope 0.10 $/min; y-intercept $30. The starting value is $30 even if you make zero calls.

8 questions on what slope and intercept mean in real-world contexts. Instant feedback after each answer.

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