In the terminology of this article, a linear function multiplies its input by a fixed constant and has no added term. Its graph is a line through the origin. Many English-language textbooks also use linear function for a line with a nonzero intercept. Here the latter is called affine so the distinction between a fixed starting value and direct proportionality stays explicit.
The slope gives the change per unit
A positive coefficient makes the graph increase. A negative coefficient makes it decrease. A zero coefficient gives the constant zero function. From two distinct inputs, find the slope by dividing the output change by the input change. Because the line passes through the origin, the output divided by the input also gives the coefficient at every nonzero input.
Find the cost of seven metres from the cost of three
If 3 metres of fabric cost 18 euros at a fixed price per metre, divide 18 by 3 to obtain 6 euros per metre. Multiply by 7 to get 42 euros. The algebraic rule extends over the real numbers, while this purchase model uses only nonnegative lengths.
A separate delivery fee of 4 euros would make the cost 46 euros for 7 metres. The fee changes the intercept, not the rate of 6 euros per metre. The new graph starts at 4 when the length is zero, so the total cost is no longer directly proportional to the length.
