For the linear function f, f(2) = 11 and f(5) = 20. What is the value of f(9)?
- Slope: m = (20 − 11)/(5 − 2) = 9/3 = 3.
- Find b from f(2) = 11: 3(2) + b = 11, so b = 5.
- So f(x) = 3x + 5, and f(9) = 3(9) + 5 = 27 + 5 = 32.
Answer: f(9) = 32
Linear functions
Digital SAT · Math · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A linear function changes by the same amount for every 1 unit increase in x. It is written f(x) = mx + b. Its graph is a straight line.
The slope m is the rate of change: how much the output rises or falls for each extra unit of input. The y-intercept b is the output when the input is 0, which is the starting value in most contexts.
Many questions give a model and ask what a number in it means. Decide whether the number is multiplied by the variable (a rate, with units such as dollars per hour) or stands alone (a starting amount). To build a function from a table or two points, find the slope first, then b.
Rules to remember
- f(x) = mx + b, where m is the slope and b is the y-intercept.
- slope = change in output ÷ change in input: m = (y2 − y1)/(x2 − x1)
- b = f(0), the value when the input is 0.
- The x-intercept is where f(x) = 0.
- A positive slope means increasing; a negative slope means decreasing.
- In a table of a linear function, equal steps in x give equal steps in f(x).
- The unit of the slope is the output unit per input unit.
Common mistake
Mixing up the slope and the intercept in a context. The number multiplied by the variable is the change per unit; the number standing alone is the value at the start.
Worked example
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