Reference · module 6
Linear functions
Everything this module explains, on one page.
What a function is
Read function notation and find values from a rule.
- A function is a rule that turns each input into exactly one output. Double it and add one turns 3 into 7 and 10 into 21.
- The rule is written f(x) = 2x + 1. f(3) means put 3 in place of x: f(3) = 2 × 3 + 1 = 7. The brackets here do not mean multiplication.
- A table lists inputs and outputs side by side. For f(x) = 2x + 1: x = 0, 1, 2 gives 1, 3, 5. Each step in x adds the same 2 to the output.
The mistake you are about to make
f(3) = 3f
f(3) = 2 × 3 + 1
Brackets after a letter usually mean multiply, so f(3) looks like f times 3. Here f is the rule's name, and f(3) means apply it to 3.
Slope and intercept
Read the slope and starting value of y = kx + b.
- A linear function is y = kx + b. Its graph is a straight line. b is where the line crosses the y-axis — the value of y when x is 0.
- k is the slope: how much y changes when x grows by 1. In y = 3x + 2, every step right raises the line by 3. A bigger k means a steeper line.
- A negative k makes the line go down: y = −2x + 8 loses 2 for every step right. With k = 0 the line is flat, and y stays equal to b.
The mistake you are about to make
y = 3x + 2: slope 2
y = 3x + 2: slope 3
The slope sounds like something added on at the end. It is the number multiplying x; the added 2 only shifts the line up.
Functions in real life
Build a linear function from a situation and use it.
- Many costs are a fixed part plus a rate: a gym at 20 euros to join plus 15 a month is C = 20 + 15m. The rate is the slope, the fixed part is the intercept.
- To build the rule, ask what is paid once and what is paid per unit. A phone at 5 euros a month plus 0.1 per minute: C = 5 + 0.1t.
- The rule answers two questions: the cost for a given amount, by substituting, and the amount for a given cost, by solving an equation.
The mistake you are about to make
C = 15 + 20m
C = 20 + 15m
The first number in the story seems to be the one multiplied. The one paid every month is the rate and goes with m.