Linear vs. exponential growth: how to tell them apart

Some quantities grow by the same amount each step. Others grow by the same percent each step. The first kind is linear, the second is exponential, and the difference matters enormously: over a long enough stretch, an exponential function always overtakes a linear one, no matter how the two start out. Algebra I asks you to tell the two apart, build a function for each, and compare them.

The one question to ask

Look at how the output changes when the input goes up by one equal step.

• If the output changes by the same amount each time (add or subtract the same number), the model is linear: f(x) = mx + b.
• If the output changes by the same factor each time (multiply by the same number), the model is exponential: f(x) = a · bˣ.

In other words, linear functions grow by equal differences over equal intervals, and exponential functions grow by equal factors over equal intervals. That single idea is the whole test.

Reading a table

Take these two tables, both with x going up by 1.

Row A: 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3. The differences are equal, so A is linear with slope 3 and starting value 5: A(x) = 3x + 5.

Row B: the differences are 3, 6, 12, which are not equal. Try ratios instead: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The ratios are equal, so B is exponential with starting value 3 and growth factor 2: B(x) = 3 · 2ˣ.

Growth rates and growth factors

A constant percent rate becomes a factor. Growing by 5% each period means multiplying by 1 + 0.05 = 1.05. Shrinking by 15% each period means multiplying by 1 − 0.15 = 0.85. A factor above 1 is growth; a factor between 0 and 1 is decay.

Worked example: decay. An 80 mg dose of a medicine loses 15% of what remains each hour. The amount after t hours is 80 · (0.85)ᵗ. After 3 hours: 0.85³ = 0.614125, so 80 × 0.614125 ≈ 49.1 mg. Notice the drop is not 15% of 80 three times (that would be 36 mg gone, leaving 44 mg). Each hour takes 15% of a smaller amount.

Building a function from two points

Worked example. A function passes through (1, 6) and (3, 54). Find both a linear and an exponential function through these points.

• Linear: slope = (54 − 6) ÷ (3 − 1) = 24. Then 6 = 24(1) + b gives b = −18, so f(x) = 24x − 18. Check: 24(3) − 18 = 54.
• Exponential: from x = 1 to x = 3 the output is multiplied by 54 ÷ 6 = 9 over two steps, so b² = 9 and b = 3 (taking the positive value). Then 6 = a · 3¹ gives a = 2, so g(x) = 2 · 3ˣ. Check: 2 · 27 = 54.

Two points fit either model; you decide from the situation which one makes sense. A constant amount added per step points to linear. A constant percent points to exponential.

Why exponential eventually wins

Worked example. Account L starts at $200 and gains $15 every week: L(w) = 200 + 15w. Account E starts at $200 and grows 5% every week: E(w) = 200 · 1.05ʷ.

The linear account is ahead for 16 weeks. At week 17 the exponential account passes it and the gap keeps widening. The linear account adds the same $15 every week, while the exponential account adds 5% of an ever larger balance. That is the heart of F-LE.3: a quantity growing exponentially eventually exceeds one growing linearly, however modest the percent and however large the fixed amount.

Common mistakes

• Testing only one pair. Check every consecutive pair, with equal steps in x. A table whose x-values skip around (0, 1, 3, 4) needs adjusting before you compare differences or ratios.
• Calling 5% growth a factor of 0.05. Growth of 5% is a factor of 1.05; 0.05 would shrink the amount by 95%.
• Treating a percent decrease as subtracting the same amount. A 15% drop each hour removes less each time. It is exponential decay, not a line.
• Forgetting the starting value. In a · bˣ, the a is the value at x = 0, not the first value in your table unless the table starts at 0.

Practice Algebra I questions with an explanation for every answer, or read the guide to California Algebra I practice.

Source: the California Common Core State Standards for Mathematics, F-LE.1, F-LE.2, F-LE.3.