An exponential graph has the form y = aˣ, where a is a positive constant and x is the power. These curves never cross the x-axis, pass through (0, 1), and either rise steeply for a > 1 or fall towards zero for 0 < a < 1. At GCSE you need to sketch, read, and interpret them.
What makes an exponential graph different from other curves?
With a quadratic (y = x²) or cubic (y = x³), the variable x is the base and the power is fixed. With an exponential, the roles are reversed: the base a is fixed and the variable x is the power.
This reversal makes exponential functions grow or decay much faster than any polynomial. Compare:
| x | y = x² | y = 2ˣ |
|---|---|---|
| 0 | 0 | 1 |
| 2 | 4 | 4 |
| 5 | 25 | 32 |
| 10 | 100 | 1024 |
| 20 | 400 | 1 048 576 |
For large x, 2ˣ dwarfs x². This rapid growth is why exponential models describe things like population growth, compound interest, and viral spread.
What are the key features of y = aˣ?
All exponential graphs y = aˣ (with a > 0, a ≠ 1) share these features:
- Passes through (0, 1): because a⁰ = 1 for any base a.
- Passes through (1, a): because a¹ = a.
- Always positive: aˣ > 0 for all x, regardless of the sign of x.
- Asymptote at y = 0: as x → −∞ (for a > 1), the curve approaches the x-axis but never touches it.
- Never crosses the x-axis: because y is always positive.
What is the difference between growth and decay?
The shape of the curve depends on whether a is greater or less than 1:
-
a > 1: the graph rises steeply to the right. This models exponential growth. Example: y = 2ˣ, y = 3ˣ, y = 1.05ˣ
-
0 < a < 1: the graph falls towards the x-axis as x increases. This models exponential decay. Example: y = (0.5)ˣ, y = (0.8)ˣ
Note that y = (0.5)ˣ and y = 2⁻ˣ are the same curve — a decay curve with base a is the reflection of the corresponding growth curve in the y-axis.
| Feature | Growth (a > 1) | Decay (0 < a < 1) |
|---|---|---|
| Shape | Rising curve, steeper to right | Falling curve, flatter to right |
| As x → ∞ | y → ∞ | y → 0 |
| As x → −∞ | y → 0 | y → ∞ |
| Asymptote | y = 0 (on left) | y = 0 (on right) |
How do you sketch an exponential graph?
Worked example: Sketch y = 3ˣ for −2 ≤ x ≤ 3.
- Build a table of values:
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | 1/9 ≈ 0.11 | 1/3 ≈ 0.33 | 1 | 3 | 9 | 27 |
- Plot the points, noting that the curve passes through (0, 1) and (1, 3).
- Draw a smooth curve that rises steeply to the right and flattens towards the x-axis on the left.
- Label the asymptote y = 0.
Key labelling requirement: always label the y-intercept (0, 1) — this is the most important single point on an exponential graph and marks are awarded for it.
How are exponential graphs used in real-life contexts?
GCSE questions often set exponential graphs in context. The standard models are:
- Compound interest / growth: V = P × aⁿ, where a = 1 + r/100 (r% growth rate), n = number of years.
- Radioactive decay / depreciation: V = P × aⁿ, where a = 1 − r/100 (r% decay rate).
Worked example: A car is bought for £12 000. Its value decreases by 15% per year. Write a formula and find the value after 5 years.
V = 12 000 × (0.85)ⁿ
After 5 years: V = 12 000 × (0.85)⁵ = 12 000 × 0.4437 ≈ £5 324
The base 0.85 gives a decay curve. Plotting V against n gives a decreasing exponential shape.
How do you read values from an exponential graph?
GCSE questions may ask you to read off x when given y, or y when given x. Use the graph carefully:
- Draw a horizontal line from the y-axis to the curve.
- Drop a vertical line to the x-axis.
- Read the x-value.
Because exponential graphs are curved (not straight), you cannot interpolate linearly. Read directly from the plotted curve, not by arithmetic.
Frequently asked questions
Why does y = aˣ always pass through (0, 1)?
Any nonzero number raised to the power zero equals 1: a⁰ = 1. This is a fundamental index law. No matter which base a you choose (as long as a > 0 and a ≠ 1), the graph always crosses the y-axis at y = 1. This is the single most reliable feature to use when sketching or checking your graph.
What happens when a = 1?
y = 1ˣ = 1 for all x — this is just the horizontal line y = 1, not a curve. That is why we require a ≠ 1 for a proper exponential. In practice, GCSE questions use bases like 2, 3, 0.5, or 0.8.
Can exponential graphs appear as transformations?
Yes — at GCSE Higher, y = 3ˣ + 2 is a translation of y = 3ˣ up by 2, changing the asymptote from y = 0 to y = 2. Similarly, y = 3^(x−1) is a translation right by 1. These are applications of graph transformation rules to exponential functions.
How do I tell an exponential graph from a quadratic on an exam paper?
Look at the y-intercept: a quadratic y = x² passes through the origin (0, 0), while an exponential y = aˣ passes through (0, 1). Also, a quadratic is symmetric (U-shape), while an exponential is not symmetric — it is asymptotic on one side. If you see a curve approaching the x-axis but never touching, it is almost certainly exponential.
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