Read the rate of change.

In y = mx + b, m is the change in y for one unit of x, and b is the value when x = 0. Calculate gradient using the same subtraction order in the numerator and denominator.

A straight-line model assumes a constant rate. Read units from the axes: if cost is measured in dollars and distance in kilometres, the gradient is dollars per kilometre. An intercept may represent a fixed charge; an extrapolation outside the observed range needs justification.

INTERACTIVE MODEL

Try it. Watch it change.

y = 1x + 0y = −x + 2

Vertical scale / gradient a1 Intercept b0 Input x1

f(1) = 1

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Move the gradient and intercept sliders. Identify which change tilts the line and which shifts it.

Gradient

m=\frac{y_2-y_1}{x_2-x_1}

Point–gradient form

y-y_1=m(x-x_1)

WORKED EXAMPLE

A taxi charges $8 initially and $2.50 per kilometre. Find the cost of 12 km.

C = 8 + 2.5d.
C(12) = 8 + 30 = $38.

Assumed knowledge

Coordinates, signed numbers and substitution.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Connect gradient and intercept to a context.
Build a linear rule from two points.

Year 10 preparation ↗

READY TO TRY IT?

Read the rate of change.

A taxi charges $8 initially and $2.50 per kilometre. Find the cost of 12 km.

Put the idea to work.