Coordinates, lines and gradients

Coordinates have an order

The point (3,-2) is three units right of the origin and two down. The point (-2,3) is two left and three up. Swapping the coordinates usually gives a different point. Use the horizontal coordinate first.

For y = 4-2x, a table gives (0,4), (1,2), (2,0) and (3,-2). Plot these pairs and draw the straight line through them. The equation describes every point on the line over the stated domain, not just the table entries.

Gradient is a ratio of changes

Using (0,4) and (2,0), the gradient is Δ y/Δ x = (0-4)/(2-0) = -2. As x increases by 1, y falls by 2. Use the same point order in numerator and denominator. Reversing both gives the same ratio; reversing only one changes its sign incorrectly.

Read the axis scales. The apparent angle of a line on a stretched screen is not its numerical gradient. A horizontal line has gradient zero. A vertical line has no finite gradient because its horizontal change is zero.

Interpret the intercept

For y = mx + c, the point at x = 0 is (0,c), so c is the y-intercept. The x-intercept is found by setting y = 0 instead. Here the y-intercept is 4 and the x-intercept is 2.

Check your understanding 1 Use coordinate differences

A line passes through (−1, 5) and (3, −3). Find its gradient and explain its sign.

Your answer or drawing on paper

Show answer

m = (-3-5)/(3-(-1)) = -8/4 = -2. The negative gradient means y decreases as x increases. The horizontal difference is 4, not 2.

A graph has a context and a domain

A cost model is C = 4 + 2d for 0 ≤ d ≤ 10 km. Its y-intercept is a fixed $4 charge and its gradient is $2 per km. Extending the mathematical line beyond the given interval does not prove the pricing model applies there.