A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Exponential Functions and Their Graphs — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 2 July 2026

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

What each mark is really for — in plain English — and the wording trap that loses it.

S1
1. **Domain:** The domain of y = 2 x is all real numbers, i.e., x ∈ ℝ .
S2
2. **Range:** The range of y = 2 x is all positive real numbers, i.e., y > 0 .
S3
3. **Intercept:** The graph passes through the point (0, 1) because 2 0 = 1 .
S4
4. **Asymptote:** The x-axis (y = 0) is a horizontal asymptote. As x → -∞ , y → 0 .
S5
5. **Growth:** Since the base 2 > 1 , the function represents exponential growth.

Model answer

Worked through, with each step tagged to the mark it earns.

S1
1. **Domain:** The domain of y = 2 x is all real numbers, i.e., x ∈ ℝ .
S2
2. **Range:** The range of y = 2 x is all positive real numbers, i.e., y > 0 .
S3
3. **Intercept:** The graph passes through the point (0, 1) because 2 0 = 1 .
S4
4. **Asymptote:** The x-axis (y = 0) is a horizontal asymptote. As x → -∞ , y → 0 .
S5
5. **Growth:** Since the base 2 > 1 , the function represents exponential growth.
Final answer: The graph of y = 2 x is an increasing curve that passes through (0, 1) and approaches the x-axis as x → -∞ .

Common mistakes

Forgetting that the range of a x is all positive real numbers. — 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. — 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. — 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. — 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 . — 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. — 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. — 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

Full worked solution (all marking points)5 marks