Same Math › Algebra 1 · grades 8–10 › Linear vs. exponential tables
How do you tell linear from exponential?
Look at how a table changes from row to row. If y goes up (or down) by the same amount each time, it's linear: 2, 5, 8, 11 adds 3. If y is multiplied by the same number each time, it's exponential: 3, 6, 12, 24 doubles. Linear rules look like y = mx + b; exponential rules look like y = a · bˣ.
- Linear: the same difference each step (common difference).
- Exponential: the same ratio each step (common ratio).
- Check every step, not just the first one.
Why schools teach it
Telling the two kinds of growth apart
Algebra 1 students tell linear from exponential situations and write both kinds of rules from tables (HSF.LE.A.1 and A.2). Savings with interest, populations and phone-battery drain are all on the test.
Exponential growth starts slow and then explodes. Seeing that in a table is the point: after enough steps, any exponential passes any line.
At the kitchen table
What to say, and what to skip
Say this
What do you add to get from one y to the next?
Is it the same every time?What do you multiply by?
Is that the same every time?What's y when x is 0?
That's the starting value: b or a.What would the next row be?
Skip this
- Deciding from the first two rows.1, 2, 4 could be doubling or adding 1, then 2. Check all rows.
- Calling any fast growth exponential.Adding 1,000 each step is still linear.
- Writing y = 2x for doubling.Doubling from 3 is y = 3 · 2ˣ.
See it in action
Reading x: 0, 1, 2, 3 and y: 80, 40, 20, 10
Differences: −40, −20, −10
not the same, so not linearRatios: 40 ÷ 80 = 1/2 each time
exponential decayStart at 80, multiply by 1/2
y = 80 · (1/2)ˣ
A ratio below 1 means it shrinks: exponential decay.
Free practice sheet
Make a linear or exponential sheet
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Parents also ask
What if it's neither?
Some tables, like 1, 4, 9, 16 (squares), have neither a common difference nor a common ratio.
What is bˣ?
b multiplied by itself x times. 2³ = 8, so y = 3 · 2³ = 24.
What's exponential decay?
Multiplying by a number between 0 and 1, like halving. It shrinks but never reaches zero.
Why does exponential always win eventually?
Each step multiplies, so the jumps keep getting bigger, while a line adds the same jump forever.