AQA A-Level Mathematics Paper 1, June 2025: Question 7

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

7 Sketches an exponential graph

which does not cross the x-axis.

Graph must be in 1st and 2nd

quadrants.

1.1a M1

Condone the correct shape

reflected in y-axis.

Condone minor errors at the

extremes.

Sketches fully correct graph with

1.1b A1

y-intercept labelled as 1 or (0,1)

Question 7 Total 2

How to answer it

AQA A-Level Mathematics • Pure Maths

Sketching Exponential Decay: y = aˣ (0 < a < 1)

What this question tests

Recognition of exponential functions in the form y = aˣ .
Understanding the geometric effect of the base constraint: 0 < a < 1 corresponds to exponential decay rather than exponential growth.
Correct curve characteristics: strictly decreasing, entirely above the x-axis (quadrants 1 and 2), with the positive x-axis acting as a horizontal asymptote.
Accurate identification and labelling of key features: specifically the y-intercept at (0, 1) .

Question 7 (2 Marks)

Sketch the curve y = aˣ on the provided axes, where 0 < a < 1

💡 Key Knowledge: Why the curve decays

When a > 1 (e.g. 2), y = 2ˣ represents exponential growth (rises from left to right).
When 0 < a < 1 , we can think of a = 1/b (where b > 1 ). Thus:
y = (1/b)ˣ = b⁻ˣ .
This is a reflection of y = bˣ in the y-axis.
As x → ∞ , aˣ → 0 (approaches y = 0 without touching or crossing it).
As x → -∞ , aˣ → ∞ .
At x = 0 , y = a⁰ = 1 for any non-zero value of a .

✅ Mark Scheme Breakdown

M1 (Method - AO 1.1a):

Sketches a smooth exponential curve in the 1st and 2nd quadrants that does not touch or cross the x-axis.
Note: Examiners condone the standard growth curve (reflection in y-axis) for M1 only, as well as minor errors at the extreme ends.

A1 (Accuracy - AO 1.1b):

Fully correct decay shape (decreasing from top left to bottom right).
The y-intercept is clearly marked and labelled as 1 or (0, 1) .

🧠 Exam Technique: 3-Step Sketch Checklist

Check the base: Always check whether a > 1 (growth) or 0 < a < 1 (decay). Here it decays.
Locate key intercepts: Set x = 0 ⇒ y = 1 . Always mark 1 clearly on the y-axis.
Draw the asymptote cleanly: As the curve extends to the right, make sure it approaches the positive x-axis asymptotically without:
- Touching the x-axis
- Crossing the x-axis
- Curling back upwards ("hooking")

❌ Common Traps & Lost Marks

Drawing exponential growth: Overlooking the condition 0 < a < 1 and sketching a typical 2ˣ or eˣ graph. This loses the accuracy mark A1 .
The "U-Turn / Hook": Curling the curve upwards as x increases. The line must continue getting closer to the x-axis.
Touching or crossing the x-axis: Exponential functions have a horizontal asymptote at y = 0 . Crossing into quadrant 4 forfeits both marks.
Missing intercept label: Failing to write 1 or (0, 1) where the graph intersects the vertical axis.

Examiner Insight: This was a quick 2-mark question intended to test fundamental function literacy. Candidates who read the condition 0 < a < 1 carefully secured 2/2 in under 30 seconds. The most common pitfall was auto-piloting into drawing standard exponential growth ( a > 1 ).

Topics

Pure Mathematics · F: Exponentials and logarithms