Mathematics from zero
Linear functions
A taxi charges 3 coins the moment you sit down, then 2 more coins for every kilometre. One kilometre: 5 coins. Two: 7. Three: 9. The cost climbs by the same step each time — that steady, predictable climb is a linear function.
After this lesson you can say what makes a function linear, read its two parts — the rate and the starting value — from its formula, and compute its output for any input.
A linear function changes by the same amount for every step of the input. Increase the input by 1 and the output rises (or falls) by a fixed amount — always the same amount. The taxi adds exactly 2 coins per kilometre, never 2 then 3. That constant step is what makes a function linear.
A linear function has the form: rate times input, plus a starting value. The taxi
is cost = 2 × n + 3, where n is the kilometres. The 2 is the steady step; the 3
is what you pay before moving at all. Every linear function fits this shape: a number
multiplied by the input, plus another number.
The rate is the slope; the starting value is the output at input 0. The number
multiplied by the input is the slope — how much the output moves per 1 step of
input. In 2n + 3 the slope is 2. The other number is the starting value — the
output when the input is 0. In 2n + 3 it is 3: at 0 kilometres the cost is already
3 coins.
A steady rate produces a straight line. The points above — (0,3), (1,5),
(2,7) and on — all lie on one straight line. They must: each step right moves the
same distance up, so the points never bend. This is exactly why the function is called
linear — its graph is a line. The next lesson looks at that graph closely.
A taxi costs 2n + 3 coins for n kilometres. Find the cost of a 4-kilometre
ride.
The function is linear: slope 2 (coins per kilometre), starting value 3 (coins
just to begin).
Substitute the input n = 4. First the rate part: 2 × 4 = 8. Then add the starting
value: 8 + 3 = 11.
The 4-kilometre ride costs 11 coins. Check the pattern: 3 km cost 9, and one more
kilometre adds exactly the slope, 2 — giving 11. The steady step holds.
Why this works
Why is the starting value the output at input 0? Because at input 0 the rate part
contributes nothing: 2 × 0 = 0. Whatever the output is at that moment comes entirely
from the other number. So the starting value is literally where the function begins
before the input has done any work — the meter reading before the taxi moves.
Common mistake
A common mistake is mixing up the two numbers — treating the starting value as the
rate. In 2n + 3, the number attached to the input n is the rate (2); the number
standing alone is the starting value (3). The rate multiplies the input; the starting
value just sits there. Check which number touches the variable.
A taxi costs 2n + 3 coins for n kilometres. Find the cost for n = 5. Type it.
In the linear function y = 4x + 1, what is the slope? Type it.
In the linear function y = 4x + 1, what is the starting value? Type it.
For the linear function y = 3x, find the output at x = 6. Type it.
For the linear function y = 2x + 1, find the output at x = 0. Type it.
What makes a function linear?
A linear function changes by the same fixed amount for every step of the input. It has the form: a rate times the input, plus a starting value. The rate — the number multiplied by the input — is the slope, how much the output moves per step. The starting value is the output when the input is 0. Because the step is constant, a linear function’s points fall on a straight line.