Graphs: including 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 only)

The idea

This Higher tier point covers exponential graphs y = kˣ for positive k, and trigonometric graphs y = sin x, y = cos x and y = tan x in degrees. For y = kˣ with k > 1, the graph rises, passes through (0, 1) since k⁰ = 1, and has the x-axis as a horizontal asymptote. For 0 < k < 1, the graph falls but still passes through (0, 1). The trigonometric graphs are periodic: y = sin x and y = cos x have a period of 360° and range from −1 to 1. The graph of y = tan x has a period of 180° and is undefined at 90°, 270°, etc., where it has vertical asymptotes. You must recognise these shapes, state key features, and use them to solve simple trigonometric equations for angles of any size, including negative angles.

What earns the marks

• Recognises that y = kˣ for k > 0 and k ≠ 1 passes through (0, 1) and has the x-axis as a horizontal asymptote; when k = 1 it is the constant line y = 1.
• Describes the shape of y = kˣ: increasing for k > 1, decreasing for 0 < k < 1, and never touching the x-axis; handles k = 1 as y = 1.
• States that y = sin x and y = cos x have period 360° and range −1 ≤ y ≤ 1, and sketches their wave shapes.
• States that y = tan x has period 180°, is undefined at 90° + 180°n, and has vertical asymptotes at those values.
• Uses the graphs to find all solutions in a given interval for equations such as sin x = 0.5 or tan x = 1, including angles of any size.

Common mistakes and how to put them right

• Thinking that y = kˣ can be zero or negative.

  Put it right: For positive k, kˣ is always positive; for k ≠ 1 the graph approaches but never touches or crosses the x-axis, while k = 1 gives y = 1.

• Believing that y = tan x is defined at 90°.

  Put it right: Tan 90° is undefined, so the graph has a vertical asymptote there, not a point.

• Assuming y = sin x and y = cos x have period 180°.

  Put it right: Their period is 360°, while y = tan x has period 180°.

Exam tips

• When sketching trigonometric graphs, mark the key points such as maximum, minimum and intercepts, and label the period clearly.
• For exponential graphs, always mark the y-intercept at (0, 1) and draw the horizontal asymptote along the x-axis.
• When solving trigonometric equations, use the graph or the unit circle to find all solutions in the required interval, and remember to include negative angles if the interval extends below 0°.