Linear functions and slope — Mathematics, 14–17 years
How a constant rate of change becomes a line on a graph, and how the line helps us predict values.
What slope tells you
A linear function changes by the same amount whenever its input increases by one unit. In y = mx + b, m is the slope: it tells how much y rises or falls, while b tells where the line crosses the y-axis. A graph makes this steady change visible.
Why use a line?
Many situations involve a fixed starting amount and a steady rate: a taxi fare, a phone plan or distance travelled at constant speed. The line model solves the problem of making predictions without listing every case. It also lets us compare rates by comparing slopes.
A taxi fare
A taxi charges €4 to start and €1.80 per kilometre. First identify b = 4 and m = 1.80, so the rule is C = 1.80d + 4. For 7 km, C = 1.80 × 7 + 4 = 12.60 + 4 = €16.60. The slope is the cost of each extra kilometre.
The common mix-up
A common mistake is to swap the slope and the starting value, writing y = bx + m. This seems reasonable because both letters are just numbers, and both affect the graph. Check their meanings instead: the value at x = 0 is the intercept, while the change between two equal steps is the slope.
Reading real rates
Linear models appear when a quantity changes at an approximately steady rate, such as fuel used over a fixed-speed journey or money earned per hour. They are useful for a first estimate, but not every process stays linear: interest, population growth and cooling often change their rate.
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