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ScientistGrades 9–11⏱ 15 min

Exponential growth

Race adding against multiplying. See when exponential growth pulls ahead, how fast things double, and how decay halves.

Standards: HSF-LE.A.1HSF-LE.A.3HSF-LE.A.2

1Wonder

Would you rather get $1,000 every day for a month, or 1 cent on day 1 that doubles every day? The answer surprises almost everyone.

2Explore

Growth race

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

01K2K3K4K012345678910Step nStep 10Exponential2 × 210= 2,048Linear2 + 10 × 10 = 102Exponential pulls ahead at step 5Doubles every step+100% each stepSillyChili
Try this
  • Press Play. At first linear growth is ahead. At which step does the exponential pull ahead?
  • Make the linear growth add 50 each step. Does exponential still win if you show more steps?
  • Set b to 1.1. That is +10% each step. How many steps does it take to double?
  • Set b below 1. What happens to the bars now?

3Explain

258111417+32, 5, 8, 11, 14, 17SillyChili
Step 1 of 5

Linear growth adds

Linear growth adds the same amount every step: 2, 5, 8, 11… (+3 each time). Its graph is a straight line.

Linear vs exponentialy=a+d ny=a⋅b n\htmlClass{c-blue}{y = a + d\,n} \qquad \htmlClass{c-purple}{y = a \cdot b^{\,n}}

Linear growth adds the same amount each step. Exponential growth multiplies by the same factor, so it eventually beats any linear growth.

4Practice

Question 1 of 5

Which sequence is exponential?

5Recap

  • Linear: add the same amount each step; a straight-line graph.
  • Exponential: multiply by the same factor b each step; y = a · bⁿ.
  • Exponential growth eventually overtakes any linear growth.
  • +r% each step means b = 1 + r/100; decay means 0 < b < 1.