Algebra - Graphs: recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, exponential functions y = kˣ for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size (Higher tier)

The idea

This is Higher tier only, and it adds two families to the shapes already covered. An exponential curve y = kˣ, with k positive, always passes through (0, 1), because a positive number raised to the power 0 is 1. When k is greater than 1 the curve climbs steeply to the right; when k is between 0 and 1 it falls away to the right. Either way it stays above the x axis and approaches it without touching. The trigonometric graphs repeat themselves: y = sin x and y = cos x both wave between −1 and 1 with a period of 360°, sine starting at 0 and cosine starting at 1. The tangent graph has a period of 180°, and it rises without limit as the angle approaches 90°, so it is drawn as separate branches with vertical asymptotes at 90° and every 180° after that.

What earns the marks

• The correct general shape of the named curve.
• The curve passing through its key point, such as (0, 1) on an exponential graph.
• The right period on a trigonometric sketch, drawn across the whole range of angles asked for.
• A value or an angle read off the curve to the accuracy the grid allows.

Common mistakes and how to put them right

• Drawing an exponential curve that crosses or touches the x axis.

• Letting a sine or cosine curve rise above 1 or drop below −1.

• Giving the tangent graph the same period as the sine graph.

• Starting the cosine curve at 0 rather than at 1, which draws the sine graph twice.

Exam tips

• Mark the horizontal scale in steps of 90° before sketching a trigonometric graph, so the wave lands on the right values.
• For y = kˣ, plot the point where x = 0 first, because the curve passes through a height of 1 there for every positive k.
• When a question asks for all the solutions in a range, read every crossing off the graph, not only the first one.