Graphs: including exponential graphs (Higher tier only)

The idea

An exponential graph comes from an equation of the form y = aˣ, where a is a positive constant not equal to 1. For a > 1, the graph rises steeply as x increases and approaches the x-axis as x decreases, so the x-axis (y = 0) is a horizontal asymptote. For 0 < a < 1, the graph falls steeply as x increases and approaches the x-axis as x increases. The graph always passes through (0, 1) because a⁰ = 1 for any non-zero a. It has no x-intercept and never touches the x-axis. To sketch it, plot points such as (−1, 1/a), (0, 1), (1, a) and (2, a²), then join them with a smooth curve that flattens towards the x-axis on one side. This topic is for Higher tier only.

What earns the marks

• Recognise that an exponential graph has equation y = aˣ with a > 0 and a ≠ 1.
• State that the graph passes through (0, 1) because a⁰ = 1.
• Identify the x-axis (y = 0) as a horizontal asymptote that the curve approaches but never meets.
• Describe the shape: for a > 1 the graph increases steeply as x increases; for 0 < a < 1 it decreases steeply as x increases.
• Sketch the graph by plotting points such as (−1, 1/a), (0, 1), (1, a) and (2, a²) and joining them with a smooth curve.

Common mistakes and how to put them right

• Drawing the graph crossing the x-axis.

  Put it right: An exponential graph never touches or crosses the x-axis, because aˣ is always positive for any real x.

• Assuming the graph passes through the origin.

  Put it right: At x = 0, y = a⁰ = 1, so the y-intercept is (0, 1), not (0, 0).

• Treating the graph as a straight line.

  Put it right: Exponential growth or decay is curved, becoming steeper or flatter, not linear.

• Confusing the direction of the curve for 0 < a < 1.

  Put it right: When the base is between 0 and 1, the graph decreases as x increases, unlike the a > 1 case.

Exam tips

• Always mark the point (0, 1) on your sketch, as it is a fixed feature of every exponential graph of the form y = aˣ.
• State the equation of the asymptote, usually y = 0, to show you understand the limiting behaviour.
• Plot at least three points, including one negative x-value, to establish the correct curvature.