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Math foundations

SAT exponential functions: growth that multiplies

Linear growth adds the same amount each step; exponential growth multiplies by the same factor. See the two race on one graph, then learn to read a, b, and the growth rate straight from the equation.

SPSAT Practice Editorial8 min read

The short version

What you'll take away

  • Linear functions add a constant amount; exponential functions multiply by a constant factor.
  • In f(x) = a · bˣ, a is the starting value and b is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays.
  • A growth rate of r% gives b = 1 + r/100; a decay rate of r% gives b = 1 − r/100.
In this article

Adding versus multiplying

Compare \(\displaystyle y = 2x + 1\) and \(\displaystyle y = 2^x\). Both equal 1 when x = 0. The linear function adds 2 each time x increases by 1: 1, 3, 5, 7, 9. The exponential function doubles: 1, 2, 4, 8, 16.
Early on, the linear function is ahead. But doubling gets faster as the numbers get bigger, so the exponential function passes the linear one between x = 2 and x = 3 and then pulls away.

The study lab

The race: add 2 or double?

y = 2x + 1 (linear) and y = 2ˣ (exponential), from x = 0 to x = 5.

What to notice: At x = 4, adding 2 each step reaches 9, while doubling reaches 16. Given enough steps, an exponential growth function will eventually exceed any linear function.

Read the equation: a and b

An exponential function has the form \(\displaystyle f(x) = a \cdot b^x\). Substituting x = 0 gives \(\displaystyle f(0) = a\), so a is the starting value. The base b is the growth factor: the number you multiply by each time x increases by 1.
If b is greater than 1, the function grows. If b is between 0 and 1, it decays. Translate a rate into a factor exactly as with percents: growth of 6% per year means b = 1.06, and a decay of 15% per year means b = 0.85.
  • \(\displaystyle f(t) = 500(1.06)^t\): starts at 500 and grows 6% per period.
  • \(\displaystyle f(t) = 20{,}000(0.85)^t\): starts at 20,000 and loses 15% per period. After two periods it is 14,450.
  • \(\displaystyle f(t) = 100 \cdot 2^{t/3}\): doubles every 3 units of t, not every unit.

Spot the model from a table or a sentence

You can identify a model without graphing. In a table with equal steps in x, check the differences and the ratios of consecutive y-values.

See the skill

Which clue points to which model?

Constant differences: 5, 8, 11, 14

Linear

Each step adds 3, so f(x) = 3x + 5.

Constant ratios: 5, 15, 45, 135

Exponential

Each step multiplies by 3, so f(x) = 5 · 3ˣ.

“increases by 40 each month”

Linear

A fixed amount is added.

“increases by 4% each month”

Exponential

A fixed percent means a fixed multiplier, 1.04.

Use the visual as a reasoning aid: tie each choice to the relationship, grammar, or evidence the passage actually gives.

Your turn

Quick practice

A quantity starts at 500 and decreases by 8% each year. Which function models the quantity after t years?

Hint: What percent of the quantity remains after each year?

Choose one answer

Practice: try these before checking the answers

Identify a and b before calculating anything.
  • 1. If \(\displaystyle f(x) = 3 \cdot 2^x\), what is f(4)?
  • 2. A table shows 5, 15, 45, 135 for x = 0, 1, 2, 3. Write the function.
  • 3. A colony of 100 bacteria doubles every 3 hours. How many are there after 12 hours?
  • 4. Does \(\displaystyle y = 400(0.75)^x\) show growth or decay, and at what percent rate?

Sources & further reading

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