Exponential graphs

Exponential graphs show equations like y = 2ˣ, where the variable x is the power. When the base is bigger than 1 the curve rises more and more steeply (exponential growth); when it is between 0 and 1 the curve falls towards zero (exponential decay). Every graph of y = aˣ passes through (0, 1) and gets ever closer to the x-axis without touching it.

What does an exponential graph look like?

Here is y = 2ˣ. In the graph, x is the power and y is the value of 2 raised to it.

Every exponential graph y = aˣ (with a positive and not 1) has the same key features:

• It passes through (0, 1), because a⁰ = 1 for any base.
• It never touches the x-axis. y is always positive, however negative x gets. The x-axis, y = 0, is a horizontal asymptote.
• There is no x-intercept, and no maximum or minimum point.
• If a > 1, it increases, and the larger a is, the steeper it climbs.
• If 0 < a < 1, it decreases, falling towards the x-axis.

In AQA's GCSE Maths specification, recognising, sketching and interpreting graphs of exponential functions y = kˣ for positive values of k is part of A12. At A-level, AQA adds y = eˣ, logarithms and exponential models.

How to draw an exponential graph

Make a table of values, plot the points and join them with a smooth curve. For y = 2ˣ:

Remember that a negative power means a reciprocal: 2⁻³ = 1/2³ = 1/8 = 0.125. The curve should get very close to the x-axis on the left but never reach it, and rise steeply on the right. Never join the points with straight lines.

Exponential growth and decay graphs

Swap the base for a number between 0 and 1 and the graph turns round. y = (½)ˣ is the same as y = 2⁻ˣ, the mirror image of y = 2ˣ in the y-axis. It still passes through (0, 1), but now it falls:

Real quantities that multiply by the same factor in each time step follow these curves:

• Compound interest: £500 earning 4% a year is worth 500 × 1.04ᵗ after t years, so 500 × 1.04¹⁰ = £740.12 after 10 years.
• Population growth: bacteria that double each generation follow N = N₀ × 2ⁿ, as in binary fission.
• Depreciation: a car worth £800 that loses 25% of its value each year is worth 800 × 0.75ᵗ, so £337.50 after 3 years.

The multiplier gives it away: bigger than 1 means growth, between 0 and 1 means decay.

Example: sketch y = 3 × 2ˣ

At x = 0, y = 3 × 1 = 3, so the y-intercept moves up from (0, 1) to (0, 3). It still has the asymptote y = 0, and every value is 3 times the value of 2ˣ: at x = 1, y = 6; at x = −1, y = 1.5. Sketch the same rising shape through (0, 3).

Now try one: decide which equations give exponential graphs, then press Mark to check in the Graphing Functions course.

The graph of y = eˣ (A-level)

e is a mathematical constant, about 2.718. Its graph y = eˣ lies between y = 2ˣ and y = 3ˣ and has all the same features: through (0, 1), asymptote y = 0, always increasing. What makes it special is its gradient. AQA's specification asks you to know that the gradient of eᵏˣ is keᵏˣ, so for y = eˣ the gradient at every point equals the y-value. At (0, 1) the gradient is 1; at (2, e²) it is e², about 7.39.

Its reflection in the y-axis, y = e⁻ˣ, is the standard decay curve, used for things like radioactive decay and drug concentrations.

Transformations of exponential graphs

Changes to the equation move the curve in predictable ways. Watch the y-intercept and the asymptote, as those are what exam sketches are marked on.

Exponential and logarithmic graphs

The logarithm is the inverse of the exponential: if y = 2ˣ then x = log₂ y. So the graph of y = log₂ x is the graph of y = 2ˣ reflected in the line y = x. Every point (a, b) on one curve becomes (b, a) on the other.

• y = 2ˣ crosses the y-axis at (0, 1) and has the horizontal asymptote y = 0.
• y = log₂ x meets the x-axis at (1, 0), with the vertical asymptote x = 0. It only exists for x > 0, because no power of 2 is zero or negative.

The what is a logarithm guide explains logs from the start, and the logarithmic graph calculator plots both kinds of curve.

Using a log graph to find an exponential model (A-level)

AQA asks you to use logarithmic graphs to estimate the constants in y = kbˣ. Taking logs of both sides turns the curve into a straight line:

So plotting log y against x gives a straight line with gradient log b and intercept log k. If the line of log₁₀ y against x has gradient 0.301 and intercept 0.477, then b = 10⁰·³⁰¹ ≈ 2.00 and k = 10⁰·⁴⁷⁷ ≈ 3.00, so the model is y = 3 × 2ˣ.

Common mistakes

• Drawing the curve crossing the x-axis. y = aˣ is always positive; the curve only gets close to y = 0.
• Starting at the origin. y = aˣ passes through (0, 1), not (0, 0).
• Mixing up 2ˣ and x². x² is a parabola, symmetrical about the y-axis. 2ˣ is not symmetrical, and grows far faster for large x.
• Getting negative powers wrong. 2⁻² = ¼, not −4.
• Forgetting the asymptote moves. y = 2ˣ + 3 levels off at y = 3, not at the x-axis.

To practise graphs alongside other topics in a short timed paper, try today's GCSE Maths Higher or A-level Maths: Algebra & Functions paper.

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