1. Classify 7,11,15,19 at x=0,1,2,3.
- Differences are 4,4,4.
- The linear rule is y=7+4x.
- Check x=3: 7+12=19.
Maths · Algebra / Precalculus · suggested level; school sequences vary
Linear vs Exponential Growth: Tables, Graphs & Examples
One jar gains 5 counters each round. Another doubles. The jar that starts ahead does not necessarily stay ahead.
Start here
A linear pattern has a constant additive change over equal input intervals. An exponential pattern has a constant positive multiplier over equal intervals.
Original BananaLearn guide · 7 November 2026
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Compare equal intervals first
A constant change means something only when input intervals are equal. A table measured every day can be checked by first differences; a table measured at uneven times needs rates over those intervals.
For x=0,1,2,3, the values 12,17,22,27 add 5 each time. Values 3,6,12,24 multiply by 2. Both increase, so “the numbers go up” does not distinguish them.
Write the linear rule
A linear rule has the form y=mx+c. Here m is the additive change per unit of x and c is the value at x=0. The first table is y=5x+12. Substituting x=3 gives 27.
If the first listed row is x=1, its y-value is not automatically c. Work backwards one interval or solve for c using the stated coordinates.
Write the exponential rule
An exponential model can be written y=A·bˣ, with positive starting value A and positive base b≠1. The second table is y=3·2ˣ. At x=0 it gives 3 because 2⁰=1.
When b>1, the model grows; when 0<b<1, it decays. A repeated 20% decrease uses b=0.8, not b=−0.2. This guide uses positive quantities so that ratio comparisons remain straightforward.
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Check differences and ratios independently
For values 2,6,18,54 at consecutive x-values, differences are 4,12,36 and are not constant; ratios are all 3. For 2,6,10,14, differences are 4 but ratios vary.
A finite table can match several more complicated formulas. Classifying a pattern as linear or exponential is a model choice supported by the given rule or evidence, not proof about every possible future row.
Understand repeated percentage change
A 10% increase gives a multiplier of 1.1. Starting from 200, repeated changes produce 200,220,242,266.2. Adding 20 each time would instead produce 200,220,240,260.
The two models agree after the first increase but diverge later because the percentage base changes. State whether a rate applies to the original amount or to the latest amount.
Find a crossover without guessing from a picture
Compare L(n)=5n+12 and E(n)=3·2ⁿ for non-negative whole-number rounds. At n=3, L=27 and E=24. At n=4, L=32 and E=48. The exponential first exceeds the linear model at whole-number round 4.
This does not identify an exact real-valued intersection between rounds. If rounds are discrete, that may be all the question requires. Do not report a whole-number answer as a precise continuous crossing time.
Read the axes before judging steepness
On ordinary equally spaced axes, a non-constant linear model graphs as a straight line. Exponential growth curves upward. A truncated axis or a logarithmic axis changes the appearance.
Plot a few values with labelled scales and show the model’s allowed input range. Neither a steep line nor a large starting value alone makes a pattern exponential.
Use models within their limits
Counter jars and fictional savings rules can demonstrate the difference safely. Real populations, prices and subscriptions rarely follow one model forever. Capacity limits and changing rates matter.
For a practical record, write the interval, starting value, additive or multiplicative rule, three checked rows and a sentence about where the model could fail. This makes a comparison useful rather than just a pattern-label exercise.
See the reasoning
Worked examples and checks
2. An amount starts at 80 and loses 25% each round. Find round 3.
- The remaining fraction each round is 0.75.
- Use 80(0.75)³=33.75.
- Check sequentially: 80→60→45→33.75; subtracting 20 each time would be a different rule.
3. Compare 10n+20 with 5·2ⁿ at non-negative whole-number n.
- At n=0,1,2,3 the pairs are (20,5), (30,10), (40,20), (50,40).
- At n=4 the values are 60 and 80.
- The exponential first exceeds the linear model at whole-number round 4. Earlier listed rounds confirm that it is the first such round.
Try, explain, check
12 questions with explained answers
Try each task before opening its explanation. These are original BananaLearn questions, not official exam or past-paper questions. Where several answers can work, the explanation gives the criteria.
Classify 4,9,14,19 at equal intervals.
Check answer and reasoning
Linear
Every difference is 5.Classify 4,12,36,108.
Check answer and reasoning
Exponential
Every ratio is 3.What multiplier represents 15% growth?
Check answer and reasoning
1.15
The new amount is 115% of the old.What multiplier represents 15% decay?
Check answer and reasoning
0.85
85% remains.Find y at x=0 for y=6·3ˣ.
Check answer and reasoning
6
3⁰=1.Find y at x=2 for y=6·3ˣ.
Check answer and reasoning
54
6×9=54.Find the first difference of 20,17,14,11.
Check answer and reasoning
−3
It is a decreasing linear pattern.Can one successful ratio establish an exponential model from two rows alone?
Check answer and reasoning
No
More values or a stated rule are needed to distinguish plausible models.Does doubling every two days mean doubling every day?
Check answer and reasoning
No
The multiplier applies to a two-day interval.Starting at 50, apply 10% growth twice.
Check answer and reasoning
60.5
50×1.1²=60.5.Starting at 50, add 5 twice.
Check answer and reasoning
60
This is additive rather than repeated percentage growth.Is a straight-looking section of a small graph enough to prove linearity?
Check answer and reasoning
No
Check the rule, scales and numerical changes.
A checking record to use in your notebook
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Original BananaLearn explanations, examples, illustrations and practice. Source facts checked 7 November 2026. Suggested levels are learning guides, not a claim that every board or textbook uses the same sequence.
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