AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 14

Mark scheme

Show the mark scheme

Q Answer Mark Comments

oe

x – B1 2 correct values

10 1 2 3

B2

y 0.5 1 2 4 8

14(a)

Additional Guidance

Mark the table only

4 or 5 points plotted correctly correct or ft their table in (a)

± square

M1 2

points may be implied by graph passing

through them

Correct smooth curve 1

A1 ± square for the five correct points

Additional Guidance

14(b)

Ignore extra points plotted with different x-coordinates

Two points with the same integer x-coordinate, with the curve going through

neither, is choice for plotting of that point and therefore incorrect

Ignore the curve drawn outside (–1, 0.5) and (3, 8)

For tolerance, ± square can be vertically or horizontally

Ruled straight lines A0

How to answer it

Exponential Functions: y = 2ˣ Table & Graph

📋 What this question tests

This question assesses your ability to calculate values for an exponential function involving positive, zero, and negative indices, and to accurately plot and draw a smooth exponential curve on a given Cartesian coordinate grid.

Part (a): Completing the Table of Values

2 Marks • Objective: Calculating powers of 2 (including negative powers)

📐 Step-by-Step Calculations

Equation: y = 2^x

For x = -1:
y = 2^-1 = 1 / 2¹ = 0.5 (or ¹/₂)
For x = 1:
y = 2¹ = 2
For x = 3:
y = 2³ = 2 × 2 × 2 = 8

✅ Completed Table

x	-1	0	1	2	3
y	0.5	1	2	4	8

Note: Any equivalent form like ¹/₂ is fully accepted.

💡 Key Knowledge: Index Rules

Negative Index Rule: a^-n = 1 / aⁿ. Hence, 2^-1 = 1 / 2.
Zero Index Rule: Any non-zero base to the power of 0 equals 1 (2⁰ = 1).
Multiplicative Pattern: As x increases by 1, y doubles (×2). Going backwards, y halves (÷2).

❌ Common Errors

2^-1 = -2: Confusing a negative index with a negative result. A negative power creates a reciprocal, never a negative number!
2³ = 6: Multiplying the base by the exponent (2 × 3) instead of multiplying 2 by itself three times.

Mark Scheme Breakdown:
• B2: All 3 missing values correct ( 0.5 , 2 , 8 ).
• B1: 2 values correct.

Part (b): Drawing the Graph of y = 2ˣ

2 Marks • Objective: Plotting coordinates and drawing a smooth curve

💡 Coordinate Points to Plot

Plot the following 5 points using a sharp pencil:

(-1, 0.5): 1 unit left, half a small square above the x-axis (since 10 small squares = 1 unit vertically, 0.5 = 5 small squares).
(0, 1): On the y-axis at y = 1.
(1, 2): 1 unit right, 2 units up.
(2, 4): 2 units right, 4 units up.
(3, 8): 3 units right, 8 units up.

🧠 Exam Technique: Drawing Smooth Curves

No straight lines: DO NOT join the points with a ruler! Exponential functions are smooth curves. Ruled line segments score 0 marks for accuracy.
Single continuous sweep: Avoid "feathering" or thick, sketchy lines. Keep your wrist loose and draw one smooth stroke.
Stay within tolerance: Points must be plotted within ±0.5 small square of their true position.

❌ Common Errors on the Graph

Ruling point-to-point: Using a ruler to create a "dot-to-dot" line graph automatically loses the A1 mark.
Curling downwards: Letting the curve curl back down towards the x-axis or dip below zero at x = -1. For all real values of x, 2^x > 0.
Ambiguous plotting: Plotting two competing points at the same x-coordinate without clearly crossing one out.

📐 Grid Scale & Tolerance Check

Horizontal Scale: 10 small squares = 1 unit (1 small square = 0.1).
Vertical Scale: 5 small squares = 1 unit (1 small square = 0.2).
Point (-1, 0.5): Place this point exactly 2.5 small squares above the x-axis at x = -1.

Mark Scheme Breakdown:
• M1: 4 or 5 points plotted correctly (±0.5 square tolerance), or correct follow-through (ft) from table in part (a).
• A1: Correct smooth curve drawn from x = -1 to x = 3 passing through all 5 correct points.

Topics

Algebra · Number · 3.1.1 Structure and calculation · 3.2.2 Graphs