MATH / IGCSE

Functions, coordinates and graphs

A function gives one output for each input in its domain. Coordinate geometry connects equations with intercepts, gradients, distance and midpoint.

Learning objectives

Explain functions, coordinates and graphs using a concrete example.
Apply the worked method and justify each step.
Recognise the limitation described in the final check.

The big idea

A function gives one output for each input in its domain. Coordinate geometry connects equations with intercepts, gradients, distance and midpoint.

Linear, quadratic, reciprocal and exponential graphs have different shapes. An inverse reverses a one-to-one mapping; solving a graph intersection means finding inputs giving equal outputs.

Equations and relationships

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

Gradient for a non-vertical line.

Worked example

Find the gradient through (1,2) and (4,8).

Follow the solution

The rise is 8−2 = 6 and the run is 4−1 = 3; 6/3 = 2.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.

Open the interactive investigation

Check your understanding

1. What is the y-intercept of y = 3x − 4?

−4
3
4

Check the answer and reasoning

−4

At the vertical axis x = 0, leaving y = −4.

2. Find the gradient through (1,2) and (4,8).

Check the answer and reasoning

The rise is 8−2 = 6 and the run is 4−1 = 3; 6/3 = 2.

3. Which relation is not a function of x over all real x?

y = x²
y = 2x
x² + y² = 1

Check the answer and reasoning

x² + y² = 1

A circle can assign two y values to the same x, so it fails the single-output condition.