GCSE maths graphs: straight lines, quadratics and curves

Straight-line and quadratic graphs, explained

A straight-line graph has an equation of the form y = mx + c. The number m is the gradient, which measures how steep the line is: the change in y divided by the change in x between any two points on it. The number c is the y-intercept, where the line crosses the y-axis (when x = 0). Parallel lines have the same gradient, so y = 2x + 3 and y = 2x − 7 are parallel, while y = 2x + 3 and y = 3x + 3 meet on the y-axis but are not parallel. Pupils are expected to use this to decide whether lines are parallel.

Pupils learn to find the equation of a line in two situations. If they are given two points, they first work out the gradient and then substitute one point into y = mx + c to find c. If they are given one point and the gradient, they skip straight to substituting. The substitution step is the key idea: a point on the line must make the equation true, so its coordinates can be used to find c.

Quadratic functions, such as y = x² − 4x − 5, have U-shaped or upside-down U-shaped graphs called parabolas. Pupils identify roots (where the graph crosses the x-axis, so y = 0), the y-intercept (where x = 0) and the turning point (the lowest or highest point). They also learn to deduce roots algebraically. Factorising x² − 4x − 5 gives (x − 5)(x + 1), so the roots are x = 5 and x = −1, and the y-intercept is −5. A parabola is symmetrical, so the turning point lies halfway between the roots, at x = 2, where y = 4 − 8 − 5 = −9. That makes it a minimum at (2, −9).

Pupils recognise and sketch a few more standard shapes. A simple cubic such as y = x³ is an S-shaped curve through the origin, climbing from bottom left to top right. The reciprocal graph y = 1/x has two separate smooth branches in opposite corners of the grid. It never touches the y-axis, because x = 0 would mean dividing by zero, so the function is only defined for x ≠ 0. It never touches the x-axis either, because 1 divided by any number is never 0. A sketch should show the correct shape and the key points, such as where it crosses the axes and any turning point, rather than an accurately plotted graph.

Graphs are also used to model real situations. On a distance–time graph, the gradient is the speed and a flat section means the object is stationary. On a speed–time graph, a flat section means constant speed and a straight sloping section shows steady acceleration, with the gradient giving the acceleration. (Working out areas under graphs and gradients of curves is part of the higher content.) Pupils plot and read from graphs of non-standard functions in context, giving approximate answers, for example estimating when a rocket reaches a certain height, or seeing that the time for a fixed journey falls as the speed rises, which gives a reciprocal curve.

Words your child will hear

Gradient

How steep a line is: the change in y divided by the change in x.

y-intercept

The point where a graph crosses the y-axis. In y = mx + c it is (0, c).

Root

A value of x where the graph crosses the x-axis, so y = 0.

Turning point

The point where a curve changes from going down to going up (a minimum), or the reverse (a maximum).

Parabola

The smooth, symmetrical curve that is the graph of a quadratic function.

Reciprocal

The result of dividing 1 by a number. The graph of y = 1/x shows this relationship.

Acceleration

How quickly speed changes. On a speed–time graph it is the gradient.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

• Use the form y = mx + c to identify parallel lines; find the equation of the line through 2 given points, or through 1 point with a given gradient
• Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically
• Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions and the reciprocal function y = 1/x with x ≠ 0
• Plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Common mistakes and how to help

How to help at home

• Graph a real journey. As passengers on a car or bus trip, note the distance travelled at a few times, or time a walk to known landmarks. Plot distance against time and ask what the gradient of each section means.
• Plot a pricing rule. A taxi charges £3 plus £2 a mile. Ask your child to make a small table, plot it, and find the equation. Which part is the gradient and which is the intercept?
• Trace a thrown ball. Throw a ball gently in the garden or park and ask your child to sketch the path. The path is very close to a parabola. Where is the turning point, and where would the roots be?
• Read a graph in the news. Look at a graph in the news, such as temperature over a week, and ask your child to describe in words what is happening and where it is steepest. The skill of reading a graph carefully is the same.

Related topics

Everything in Key Stage 4 maths