SAT Exponential Functions: Growth, Decay, and Graphs

Exponential functions model situations where quantities grow or shrink by a constant percentage. They show up in SAT questions about population growth, compound interest, and radioactive decay.

The General Form

• = initial value (the y-intercept, when )
• = growth/decay factor
  • If : exponential growth
  • If : exponential decay

Growth vs. Decay

Example 1 (Growth): A bacteria population starts at 500 and doubles every hour.

After 3 hours:

Example 2 (Decay): A car worth 20{,}000$ loses 15% of its value each year.

After 3 years: 12{,}282.50$

Notice: losing 15% means keeping 85%, so .

Percent Change and the Growth Factor

If a quantity increases by per period:

If it decreases by per period:

Example 3: A town's population grows by 3% per year from 10,000.

Reading Exponential Graphs

On the SAT, you might see a graph and need to identify:

• The y-intercept () — where the curve crosses the y-axis
• Whether it shows growth (curving up) or decay (curving down)
• The growth factor from a table of values

To find from a table: divide any -value by the previous one.

So — a 20% increase per unit.

Compound Interest

The SAT uses this formula:

• = principal (starting amount)
• = annual interest rate (as a decimal)
• = number of times compounded per year
• = years

Example 4: 5{,}000$ invested at 4% compounded quarterly for 6 years.

$6{,}348.67$$

Practice Problems

Problem 1: A radioactive substance has a half-life of 5 years. If you start with 80 grams, how much remains after 15 years?

Problem 2: An investment grows according to . What is the annual growth rate?

Problem 3: Which function shows exponential decay?
(A) (B) (C) (D)

Key Takeaways

• Exponential functions have the form
• = growth; = decay
• Percent increase of means
• To find the growth factor from a table, divide consecutive -values
• Compound interest is just a specific exponential function

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