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GCSE Maths · Algebra
Exponential graphs GCSE Questions and Worked Answers
An exponential graph models a fixed multiplier for each equal input step. In y = a × bˣ, a is the value at x = 0 and b is the multiplier for one step.
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Start with the meaning
What you need to know about exponential graphs
Start with one counter and double the number at each stage. At stages 0, 1, 2 and 3 you have 1, 2, 4 and 8 counters. The increase gets larger, but the multiplier stays two. Write the stage as x and the number as y: y = 2ˣ means a product of x factors, each equal to two, when x is a positive whole number. Working backwards halves the number at each step, so the mathematical graph also has fractional values.
See the idea first
Equal steps multiply, rather than add
At x = 0 the value is 1. At x = −1 it is 1/2, and at x = −2 it is 1/4. The graph continues smoothly for real x; a counter model uses only whole stages. The horizontal axis is an asymptote: the curve approaches it but does not reach it.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Plot from an equation
What the problem asks: Plot y = 3ˣ for x = −1, 0, 1, 2.
How to solve it: Substitute each input to obtain 1/3, 1, 3, 9. Plot the pairs and draw a smooth increasing curve.
Recognise decay
What the problem asks: Explain y = (1/2)ˣ.
How to solve it: Every step right halves the height. The curve decreases but stays above zero.
Recover an equation
What the problem asks: A model y = a × bˣ has values 5, 15, 45 at x = 0, 1, 2. Find a and b.
How to solve it: The starting height gives a = 5. Successive values have ratio 3, so b = 3. The equation is y = 5 × 3ˣ.
A reliable routine
Work with a fixed-multiplier model
For y = a × bˣ here, a > 0 and b > 0. Successive heights have constant ratio b, explaining both table construction and the graph's shape.
- Find the starting value at x = 0.
- Multiply by b for each unit right and divide by b for each unit left.
- Plot a table and join smoothly; retain a positive height.
- Use 1 + r/100 for growth of r% or 1 − r/100 for decay of r%.
Check: For b > 1 the curve grows; for 0 < b < 1 it decays. For b = 1 the graph is constant, y = a. A shifted curve can have a different asymptote. No logarithms are needed for these questions.
Fully worked
Exponential graphs GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
A table of powers
Question
Find y = 3ˣ at x = −2, −1, 0, 1, 2.
Negative powers mean reciprocals.
The complete values are 1/9, 1/3, 1, 3, 9.
Example 2
Key features
Question
State the intercept and asymptote of y = 4 × 2ˣ.
At x = 0,
The y-intercept is (0, 4). Repeated halving to the left approaches zero, so the asymptote is y = 0.
Example 3
A decay curve
Question
For y = 12 × (1/2)ˣ, find the heights at x = 0, 1, 2.
Start at 12 and halve each step: 12, 6, 3. The curve decreases towards zero without crossing it.
Example 4
Find the multiplier
Question
An exponential model has y = 7 at x = 0 and y = 28 at x = 1. Find its equation.
The initial value is 7.
Therefore
Example 5
Read a simple intersection
Question
Solve 2ˣ = 8. Explain what this means on a graph.
Thus x = 3. The graph y = 2ˣ meets horizontal y = 8 at (3, 8).
Example 6
Contextual growth
Question
A model starts at 200 organisms and grows by 10% per hour. Find its value after three hours.
Each hour retains 100% and adds 10%, so the multiplier is 1.1.
The model predicts about 266 organisms. A real count is whole, while the model value need not be.
10 original questions · total 24 marks
Exponential graphs GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 29 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Substitute
Find 5².
Show worked answer
This is the height of y = 5ˣ at x = 2.
Zero input
Find the y-intercept of y = 6 × 3ˣ.
Show worked answer
Intercept: (0, 6).
Negative input
Find y when y = 2ˣ and x = −3.
Show worked answer
Recognise decay
Does y = 9 × 0.8ˣ grow or decay as x increases?
Show worked answer
It decays: each step keeps 80% of the previous value, and 0 < 0.8 < 1.
Identify a model
An exponential table has y = 4, 12, 36 at x = 0, 1, 2. Find its equation.
Show worked answer
Initial height 4; fixed ratio 12/4 = 36/12 = 3.
Not exponential
Values 3, 6, 9 occur at x = 0, 1, 2. Is this a positive exponential model?
Show worked answer
No. The ratios 6/3 = 2 and 9/6 = 1.5 differ. The constant addition of 3 suggests a linear model.
Decay calculation
A model starts at 500 and loses 20% each step. Find its value after two steps.
Show worked answer
The retained multiplier is 0.8.
A shifted curve
State the horizontal asymptote of y = 2ˣ + 3.
Show worked answer
As 2ˣ approaches zero to the left, y approaches 3. The asymptote is y = 3, not y = 0.
Fractional power
Solve 4ˣ = 2.
Show worked answer
Therefore x = 1/2.
Graph estimate
A graph of y = 2ˣ crosses y = 6 between x = 2 and x = 3. Explain why, and comment on the graph reading x ≈ 2.6.
Show worked answer
The increasing curve crosses in between. The reading x ≈ 2.6 is approximate, not an exact equality.
Examiner-style feedback
Common exponential graphs mistakes
In 2ˣ the input is the power, not a factor.
2⁻¹ is 1/2, not −2.
Only y = bˣ has intercept one; a multiplier a changes it.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Find the fixed ratio.
- Include x = 0.
- Negative powers are reciprocals.
- Distinguish exact values from graph estimates.
Quick answers
Exponential graphs FAQ
Is compound interest exponential?
Yes, when the same percentage applies each period. Our compound-interest guide focuses on the financial calculations.
Can y = 2ˣ equal zero?
No, for any finite real x it is positive.
Content standards
Curriculum and rights review
A12/A14 Higher exponential graphs; R16 fixed growth/decay models. Positive bases; no logarithms or calculus. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references