An exponential function has the variable in the power: y = aˣ, where a is a positive constant. For a > 1 the graph rises (growth); for 0 < a < 1 it falls (decay). Every graph of y = aˣ passes through (0, 1), stays above the x-axis and has the asymptote y = 0. Transformations move the intercept and, for vertical translations, the asymptote.
Key features of y = aˣ
- Passes through (0, 1), since a⁰ = 1
- y > 0 for all x
- Horizontal asymptote y = 0
- a > 1: increasing (growth). 0 < a < 1: decreasing (decay)
- (1/a)ˣ = a⁻ˣ, the reflection of aˣ in the y-axis
What is an exponential function?
In y = aˣ the variable x is the power, unlike a power function such as y = x², where x is the base. That difference gives exponential graphs their characteristic shape: very slow change on one side and very rapid change on the other.
For any base a > 1, the graph passes through (0, 1), is always above the x-axis, and gets closer and closer to y = 0 without reaching it.
Growth, decay and bases below 1
The base decides the direction. If a > 1 the graph increases, which models growth. If 0 < a < 1 it decreases, which models decay. Both pass through (0, 1) and both have the asymptote y = 0. A decay curve is a growth curve reflected in the y-axis, because (1/a)ˣ = a⁻ˣ.
Vertical shifts and stretches
y = aˣ + k moves the whole curve up by k, so the asymptote moves to y = k and the y-intercept becomes (0, 1 + k). y = kaˣ stretches the curve vertically, so the y-intercept becomes (0, k) but the asymptote stays at y = 0, because k × 0 = 0.
That difference is a favourite exam point: adding moves the asymptote; multiplying does not.
Reflections and horizontal shifts
y = −aˣ reflects the curve in the x-axis: it passes through (0, −1) and lies below the asymptote y = 0. y = a⁻ˣ reflects it in the y-axis, turning growth into decay. y = aˣ⁻ᵇ shifts it right by b. These combine with vertical shifts, so always work out the asymptote and intercept for the final equation.
Exponential models
Real growth and decay are modelled by y = Aaᵗ, where t is time. A is the initial value, the value when t = 0, and a is the multiplier per time period. An increase of 20% per period means a = 1.2; a decrease of 15% means a = 0.85. Finding the time to reach a particular value needs logarithms.
Worked examples
Example 1. For y = 3ˣ, find the y-intercept and state the asymptote.
- At x = 0: y = 3⁰ = 1.
- As x → −∞, 3ˣ → 0.
- 3 > 1, so the curve is increasing.
Answer: Intercept (0, 1), asymptote y = 0, increasing.
Marks: B1 for (0, 1), B1 for y = 0.
Example 2. Is y = (3/4)ˣ increasing or decreasing, and what is its asymptote?
- The base is 0.75, between 0 and 1.
- So the curve decreases.
Answer: Decreasing (decay), with asymptote y = 0.
Marks: B1 for decreasing, B1 for the asymptote.
Example 3. State the asymptote and y-intercept of y = 4ˣ − 2.
- The whole curve moves down 2, so the asymptote is y = −2.
- At x = 0: y = 4⁰ − 2 = −1.
Answer: Asymptote y = −2, intercept (0, −1).
Marks: B1 for the asymptote, B1 for the intercept.
Example 4. Describe the single transformation mapping y = 2ˣ to y = 2⁻ˣ.
- Only the sign of x has changed.
- That reflects the curve in the y-axis, turning growth into decay.
Answer: A reflection in the y-axis.
Marks: B1 for the reflection.
Example 5. A population is modelled by P = 500 × 1.2ᵗ, where t is in years. Find the initial population, the population after 3 years, and interpret 1.2.
- t = 0: P = 500 × 1 = 500.
- t = 3: P = 500 × 1.728 = 864.
- Multiplying by 1.2 each year is an increase of 20% a year.
Answer: 500 initially; 864 after 3 years; 20% growth each year.
Marks: B1 for 500, M1 for 500 × 1.2³, A1 for 864.
Common mistakes
- Moving the asymptote for a stretch. y = 3 × 2ˣ still has asymptote y = 0; only adding or subtracting moves it.
- Thinking the graph crosses the x-axis. aˣ is always positive, so y = aˣ never reaches 0.
- Confusing aˣ with xᵃ. In an exponential the variable is the power, not the base.
- Using the percentage as the multiplier. A 20% increase means multiplying by 1.2, not 0.2.
- Forgetting the new intercept. For y = aˣ + k, the intercept is (0, 1 + k).
How exponential graphs are examined
Exponential functions open the exponentials and logarithms chapter of Pure 2 (WMA12) in Edexcel International A-Level, and appear in first-year Pure for Edexcel A-Level (9MA0). Questions ask you to:
- sketch y = aˣ or a transformation, labelling the intercept and asymptote
- find the base from a point on the curve
- interpret A and a in a growth or decay model
- use a model to find a value at a given time
For graph and model questions with mark schemes, see the exponential and log exam questions for Pure 2. Practise further with the exponentials and logarithms worksheet or the A-Level exponentials and logarithms revision engine.
Logarithms undo exponentials: see introducing logarithms. In Pure 3, eˣ and ln x take this further.
Frequently asked questions
Why does y = aˣ never touch the x-axis?
Because a positive number raised to any power is always positive. As x becomes very negative, aˣ gets close to 0 but never reaches it.
What is the difference between exponential growth and decay?
Growth has a base greater than 1, so the graph rises. Decay has a base between 0 and 1, so the graph falls.
Which transformations move the asymptote of y = aˣ?
Adding or subtracting a constant, y = aˣ + k, moves it to y = k. Stretches, reflections in the y-axis and horizontal shifts leave it at y = 0.