Key Stage 4 maths · Algebra

GCSE higher graphs: exponential, trig, transformations, circles

This topic covers the curves and graph skills beyond straight lines and quadratics that appear at the top of GCSE maths. Pupils sketch and interpret exponential and trigonometric graphs, translate and reflect graphs, estimate gradients and areas under graphs in real contexts, and use the equation of a circle centred at the origin, including finding a tangent.

Exponential, trig and circle graphs, explained

An exponential graph has the form y = kˣ, with k a positive number. Because k⁰ = 1, every one passes through (0, 1). When k is greater than 1, the curve climbs faster and faster, as in 1, 2, 4, 8, 16 for y = 2ˣ. When k is between 0 and 1, it falls and flattens towards zero, which models decay. The curve gets closer to the x-axis but never touches it. Pupils also plot exponential graphs from real situations, such as a doubling population or savings with compound interest, and read approximate answers off the curve, for example how long something takes to reach a target.

The graphs of y = sin x, y = cos x and y = tan x are drawn with the angle x in degrees, and for angles of any size, not just up to 90°. Sine and cosine are smooth waves that repeat every 360° and stay between −1 and 1. Sine starts at 0 and cosine starts at 1. Tangent repeats every 180° and has no value at 90°, 270° and every 180° on from those, where the graph breaks. Between those breaks it climbs through every value from very large negative to very large positive. Pupils use the shapes to find values such as sin 150° = 0.5 and to list all the solutions of an equation like sin x = 0.5 between 0° and 360°, which are 30° and 150°.

Pupils also sketch translations and reflections of a graph they are given. Adding a number outside the function, y = f(x) + a, moves the graph up by a. Adding inside, y = f(x + a), moves it left by a. A minus sign outside, y = −f(x), reflects it in the x-axis, and a minus sign inside, y = f(−x), reflects it in the y-axis. A sketch does not need exact plotting, but it should show the key features, such as turning points and where the graph crosses the axes, in their new positions.

A curve’s gradient changes from point to point, so pupils estimate the gradient at a point by drawing a tangent with a ruler and measuring its rise over its run. The area under a graph is estimated by splitting it into trapezia and adding up. What the answers mean depends on the axes: on a distance-time graph the gradient is speed; on a velocity-time graph the gradient is acceleration and the area is distance travelled. The same ideas apply in financial graphs, where a gradient is a rate of growth in money. Results are estimates, and pupils are expected to say so.

The equation of a circle with its centre at the origin is x² + y² = r², where r is the radius. So x² + y² = 25 is a circle of radius 5 passing through (5, 0), (0, 5) and (3, 4). A tangent touches the circle at one point and is perpendicular to the radius at that point. That fact links circles back to gradients and perpendicular lines: find the gradient of the radius, take the negative reciprocal, and write the equation of the line.

Words your child will hear

Exponential: A relationship where the variable is in the power, such as y = 3ˣ, giving rapid growth or decay.
Asymptote: A line a curve gets closer and closer to without reaching, like the x-axis for y = 2ˣ.
Period: The distance along the x-axis before a graph starts repeating: 360° for sine and cosine.
Translation: Sliding a graph without turning or flipping it.
Tangent: A straight line that touches a curve at one point, matching the curve’s direction there.
Trapezium: A four-sided shape with one pair of parallel sides. Splitting the area under a graph into trapezia and adding their areas gives an estimate.
Radius: The distance from the centre of a circle to its edge. It appears squared in the equation.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Recognise, sketch and interpret graphs of the exponential function y = kˣ for positive values of k
Recognise, sketch and interpret graphs of the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size
Sketch translations and reflections of the graph of a given function
Plot and interpret exponential graphs in real contexts to find approximate solutions to problems
Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point

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

A worked example

Find the equation of the tangent to the circle x² + y² = 25 at the point (3, 4).

Check the point is on the circle: 3² + 4² = 9 + 16 = 25. It is.
The centre is (0, 0), so the radius to (3, 4) goes 3 across and 4 up. Its gradient is 4 ÷ 3.
A tangent is perpendicular to the radius at the point of contact, so its gradient is the negative reciprocal of 4 ÷ 3, which is −3 ÷ 4.
Use the point (3, 4) and the gradient to write the line, then multiply through by 4 to remove the fraction.
```
y - 4 = -3/4 (x - 3)
4y - 16 = -3x + 9
3x + 4y = 25
```
Check: the point (3, 4) gives 3 × 3 + 4 × 4 = 25, so it lies on the line, and the gradient is −3 ÷ 4.

The tangent is 3x + 4y = 25, which can also be written y = −¾x + 25/4.

Common mistakes and how to help

Thinking y = f(x + 2) moves the graph to the right

The plus sign suggests moving in the positive direction, so the graph is drawn 2 to the right.

What helps: Look at what makes the bracket zero. For y = (x + 2)², the lowest point is where x = −2, so the graph has moved left. Adding outside the function, y = f(x) + 2, moves it up.

Reading x² + y² = 25 as a circle with radius 25

The number on the right is the first thing seen, and pupils forget it is the radius squared.

What helps: Test the axis crossings: x = 5, y = 0 gives 25 = 25. Always take the square root to get the radius.

Thinking sine and tangent graphs stop at 90°

Right-angled triangle work only uses angles below 90°, so it is easy to assume the curves simply stop there, or that sine keeps climbing.

What helps: Sketch the full wave every time: sine rises to 1 at 90° and comes back down. That is why sin 150° equals sin 30°. Check the calculator is in degree mode.

Confusing y = 2ˣ with y = x²

Both involve a 2 and an index, and for small values they look alike: 2⁴ and 4² are both 16.

What helps: Make a table. At x = 5, 2ˣ is 32 but x² is only 25, and at x = 10 the gap is 1,024 against 100. In an exponential the variable is the power, and the curve never reaches zero.

How to help at home

Walk a distance-time graph. Time a steady walk across a garden or park, then a walk that starts slowly and speeds up. Sketch each as a graph of distance against time and talk about which has the steeper gradient.
Fold paper to see doubling. Fold a sheet in half again and again, counting layers: 1, 2, 4, 8, 16. Plot the results and notice how quickly the curve climbs. Ask how many layers 10 folds would give if real paper allowed it (1,024).
Draw a circle on squared paper. Use a pencil and a loop of string, with one person holding the centre at the origin, to draw a circle of radius 5. Then list the grid points on it, such as (3, 4) and (−4, 3), and check them in x² + y² = 25.
Look for waves. A playground swing, tides and a point on a turning wheel all repeat over time. Sketch the height of one against time and mark the period, the time for one full repeat.
Slide a sketch around. Sketch y = x² on a sheet of tracing or thin paper and slide or flip it over the original. Predict the new equation each time before checking.

Builds on

Key Stage 3: Algebraic notation and formulae
Key Stage 3: Solving linear equations
Key Stage 3: Graphs of linear and quadratic functions
Key Stage 3: Sequences and the nth term