Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
A taxi displays: pick up fee 3 euros, then 2 euros per kilometre. For a trip of x kilometres, you pay f(x) = 2x + 3 euros. This formula, where x is multiplied by one number then increased by another, defines a linear function: f(x) = ax + b.
The two numbers a and b each tell a story. The number b is the y-intercept: it is f(0), the starting point (here the 3 euros you pay before even moving). The number a is the slope: it is the speed of change (here, every kilometre adds 2 euros).
Graphically, a linear function always draws a straight line. The line crosses the vertical axis at height b, and the coefficient a gives the slope: when x increases by 1, f(x) increases by a. If a is positive, the line goes up (increasing function). If a is negative, it goes down (decreasing function). If a is zero, it is horizontal.
Two points are enough to know the whole line. The slope is computed from two points with the formula: a = (change in y) ÷ (change in x). Watch out for the classic trap: do not mix up a and b! In f(x) = 5x + 2, the slope is 5 (the number multiplying x) and the y-intercept is 2.
Describe the function f(x) = 2x + 3: starting point and rate of change.
f(0) = 2 × 0 + 3 = 3
The y-intercept is 3: the line crosses the vertical axis at height 3.
f(1) = 5, f(2) = 7, f(3) = 9
Each time x increases by 1, f(x) increases by 2: that is the slope.
a = 2 > 0
The coefficient is positive, so the line goes up: the function is increasing.
Result: f starts at 3 and climbs by 2 at each step: increasing line with slope 2.
Find the linear function f such that f(0) = 1 and f(2) = 7.
b = f(0) = 1
The y-intercept can be read directly: it is the value at x = 0.
a = (7 - 1) ÷ (2 - 0) = 6 ÷ 2 = 3
The slope is the change in y divided by the change in x.
f(x) = 3x + 1
I assemble a and b, then check: f(2) = 3 × 2 + 1 = 7, correct.
Result: f(x) = 3x + 1