SAT Exponential Growth & Decay: Complete Guide + Practice

Exponential functions are part of SAT Advanced Math.

College Board includes exponential equations and nonlinear functions within this domain. See the rest of our SAT Math guides for other Advanced Math topics.

The most important idea is:

equal steps in x create equal multiplication factors—not equal additions.

General form

[ y=a(b)^x ]

where:

• (a) = initial value;
• (b) = growth or decay factor.

Growth

If a quantity grows by 8% each period:

[ b=1+0.08=1.08 ]

So:

[ y=a(1.08)^x ]

Decay

If a quantity decreases by 8% each period:

[ b=1-0.08=0.92 ]

So:

[ y=a(0.92)^x ]

Common SAT trap

A decrease of 80% means you keep:

[ 20%=0.20 ]

So the factor is:

[ 0.20 ]

not:

[ 0.80 ]

Example

A bacteria culture starts with 500 cells and grows by 12% per hour.

[ P(t)=500(1.12)^t ]

After 3 hours:

[ P(3)=500(1.12)^3 ]

Recognizing exponential tables

Each y-value is multiplied by 3.

Exponential.

A table for a linear equation would add the same amount each step instead.

Percent change from the factor

If:

[ y=200(1.35)^x ]

growth rate:

[ 35% ]

If:

[ y=200(0.73)^x ]

decay rate:

[ 27% ]

because:

[ 1-0.73=0.27 ]

Repeated percent change is not simple addition

10% growth for 3 years is not simply 30% growth.

It compounds:

[ (1.10)^3=1.331 ]

Total growth: 33.1%.

Exponential curves are easy to mix up with the curves you see in quadratic equations, but the two behave very differently as x grows.

Original practice

1

A population grows 6% per year. Growth factor?

2

A machine loses 15% of its value each year. Decay factor?

3

What percent growth is represented by:

[ f(x)=80(1.24)^x ]

4

What percent decay is represented by:

[ g(x)=90(0.68)^x ]

5

A quantity starts at 40 and doubles each period. Equation?

Answers

1. 1.06
2. 0.85
3. 24%
4. 32%
5. (40(2)^x)

Common traps

• using percentage instead of growth factor;
• confusing decrease with amount remaining;
• treating exponential change as linear;
• adding repeated percentages instead of compounding.

For the factoring side of Advanced Math, see our guide to polynomials and factoring.

Official source

• College Board Advanced Math
  https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced

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