A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Exponential Functions and Their Graphs — mark scheme explained
The short answer
Exponential functions are a fundamental part of A-Level Mathematics, particularly in the study of growth and decay. This section focuses on understanding the function a x where a is positive, and the special case of the natural exponential function e x .
The question
Sketch the graph of y = 2 x . Identify its domain, range, and any asymptotes.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For sketching graphs, ensure you clearly label key points and features. For identifying properties, be precise and use correct mathematical terminology. When finding derivatives or integrals, show all steps and include any necessary constants.
What the command words demand
- Sketch
- Draw a graph of the function, including key features such as intercepts and asymptotes.
- Identify
- Determine specific characteristics of the function, such as domain, range, and asymptotes.
- Find
- Calculate or determine a value, derivative, or integral related to the function.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-10 minutes for questions involving exponential functions, depending on the complexity of the task.
- 1. **Domain:** The domain of y = 2 x is all real numbers, i.e., x ∈ ℝ .1 mark
- 2. **Range:** The range of y = 2 x is all positive real numbers, i.e., y > 0 .1 mark
- 3. **Intercept:** The graph passes through the point (0, 1) because 2 0 = 1 .1 mark
- 4. **Asymptote:** The x-axis (y = 0) is a horizontal asymptote. As x → -∞ , y → 0 .1 mark
- 5. **Growth:** Since the base 2 > 1 , the function represents exponential growth.1 mark
Final answer: The graph of y = 2 x is an increasing curve that passes through (0, 1) and approaches the x-axis as x → -∞ .
Work through every step correctly and you earn all 5 marks.
Another worked example
Sketch the graph of y = e x . Identify its domain, range, and any asymptotes.
- 1. **Domain:** The domain of y = e x is all real numbers, i.e., x ∈ ℝ .1 mark
- 2. **Range:** The range of y = e x is all positive real numbers, i.e., y > 0 .1 mark
- 3. **Intercept:** The graph passes through the point (0, 1) because e 0 = 1 .1 mark
- 4. **Asymptote:** The x-axis (y = 0) is a horizontal asymptote. As x → -∞ , y → 0 .1 mark
- 5. **Growth:** Since the base e > 1 , the function represents exponential growth.1 mark
Final answer: The graph of y = e x is an increasing curve that passes through (0, 1) and approaches the x-axis as x → -∞ .
Work through every step correctly and you earn all 5 marks.
Common mistakes
Forgetting that the range of a x is all positive real numbers.
Why it happens: Students sometimes confuse the range with the domain or forget that exponential functions never reach zero.
Fix: Always remember that the range of y = a x is y > 0 . This can be verified by plotting points or recalling the properties of exponential functions.
Misinterpreting the horizontal asymptote as a vertical line.
Why it happens: Students might confuse the x-axis (y = 0) with a vertical line, especially when sketching graphs quickly.
Fix: Clearly label the x-axis as the horizontal asymptote and ensure it is drawn horizontally. The graph approaches this line but never touches it.
Forgetting that e x is its own derivative and integral.
Why it happens: Students might mix up the properties of different functions or forget this unique property of the natural exponential function.
Fix: Memorize that the derivative and integral of e x are both e x . Practice problems involving differentiation and integration to reinforce this concept.
Confusing exponential growth with decay when the base is between 0 and 1.
Why it happens: Students might not fully understand the difference between a > 1 (growth) and 0 (decay).
Fix: Always check the value of the base. If a > 1 , it represents growth; if 0 , it represents decay.
Forgetting to include the constant of integration when finding the integral of e x .
Why it happens: Students might forget that indefinite integrals require a constant of integration, especially for simple functions like e x .
Fix: Always include the constant of integration (C) when finding an indefinite integral. This is crucial for ensuring the solution is complete and correct.
Misinterpreting the point (0, 1) as a y-intercept instead of an x-intercept.
Why it happens: Students might confuse the intercepts or forget that exponential functions do not have x-intercepts.
Fix: Remember that the graph of y = a x passes through (0, 1) and does not cross the x-axis. This point is a y-intercept, not an x-intercept.
Forgetting that the domain of a x is all real numbers.
Why it happens: Students might think the domain is limited to positive or non-negative values, especially when dealing with exponential growth and decay.
Fix: Always recall that the domain of y = a x is all real numbers, i.e., x ∈ ℝ . This can be verified by plotting points or recalling the properties of exponential functions.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Sketch Exponential Graph | Sketch y = 2^x and state its domain, range, and asymptote. | 5 |
| Exponential Graph Sketch | Sketch y = e^x and state its domain, range, and asymptote. | 5 |
| Differentiate Exponential Function | Find the derivative of an exponential function with respect to x. | 2 |
| Integrate Exponential Function | Integrate the exponential function e^x, showing steps and including the constant of integration. | 2 |
| Evaluate Zero Index | State the value of a number raised to the power of zero. | 1 |
| Total across these question types | 15 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.