Nonlinear functions 2: exponential growth and decay
Advanced Math is about 35% of SAT Math (13–15 of 44 questions). Exponential functions appear as stories about money, populations and radioactive decay, as tables, and as questions about what the numbers a and b mean.
The shape f(x) = a·b^x
In an exponential function the variable is in the exponent. The number a is the starting value, the value at x = 0, so the graph crosses the y-axis at (0, a). The number b is the factor: every time x goes up by 1, the output is multiplied by b. If b > 1 the function grows, and the growth gets faster and faster. If 0 < b < 1 the function decays: it falls quickly at first, then flattens out, getting close to 0 but never reaching it.
From a rate to a factor
A percent change per period turns into a factor: growth factor = 1 + rate, decay factor = 1 − rate (rate as a decimal). A quantity that grows 8% each year has b = 1.08. One that loses 15% each year has b = 0.85. The words “doubles”, “triples” and “halves” mean b = 2, b = 3 and b = 0.5. Never use the rate itself (0.08 or 8) as the factor.
| Description | Factor b | Function |
|---|---|---|
| grows 8% each year | 1.08 | a(1.08)^t |
| falls 15% each year | 0.85 | a(0.85)^t |
| doubles every year | 2 | a(2)^t |
| halves every year | 0.5 | a(0.5)^t |
| grows 100% each year | 2 | a(2)^t |
When one period is not one unit
Often the factor applies over a period that is not 1. If a quantity doubles every 3 years, then after t years it has doubled t/3 times, so f(t) = a·2^(t/3). If a substance has a half-life of 6 hours, the amount after t hours is a·(1/2)^(t/6). The exponent is always (time passed) ÷ (length of one period). If a monthly rate is given but t is in years, multiply instead: 1.02^(12t).
Linear or exponential? Look at the table
Look at the outputs when x goes up by 1. If you add the same amount each time (equal differences), the table is linear. If you multiply by the same number each time (equal ratios), it is exponential. A table can increase without being exponential. Table A adds 3 each time, so it is linear. Table B multiplies by 2 each time, so it is exponential.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Table A | 5 | 8 | 11 | 14 |
| Table B | 5 | 10 | 20 | 40 |
Compound interest
Money in a bank grows exponentially when interest is paid on the interest already earned. If you deposit P dollars at a yearly rate r (as a decimal) and interest is added once a year, the balance after t years is P(1 + r)^t. If interest is added n times a year, the balance is P(1 + r/n)^(nt). For example, $2,000 at 5% compounded annually becomes 2,000(1.05)³ = $2,315.25 after 3 years.
| Hours | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| N(t) | 400 | 1,200 | 3,600 | 10,800 |
- N(t) = 400(3)^(2t)
- N(t) = 400(3)^(t/2)
- N(t) = 400(2)^(t/3)
- N(t) = 3(400)^(t/2)
- The starting value is a = 400.
- The factor is 3, and it is applied once every 2 hours, so the exponent is t/2.
- N(t) = 400(3)^(t/2).
- Check t = 4: that is two tripling periods, and 400 · 3² = 3,600 ✓
- Decay factor: 1 − 0.12 = 0.88.
- V(t) = 18,000(0.88)^t.
- V(3) = 18,000 × 0.681472 = 12,266.496, which is about $12,266.
- The population increases by 6 people each year.
- The population increases by 6% each year.
- The population in 2020 was 5,300.
- The population in 2020 was 6% of 5,000.
- a = 5,000 is the population at t = 0, which is the year 2020.
- b = 1.06 = 1 + 0.06, so each year the population is multiplied by 1.06: a 6% increase.
- The 6% is taken from the current population, so the number of people added each year keeps growing.
- 15 ÷ 5 = 3, so the sample is halved 3 times.
- A(t) = 640(1/2)^(t/5).
- A(15) = 640 × (1/2)³ = 640 ÷ 8 = 80.
- Using the rate as the factor8% growth means b = 1.08, not 0.08 or 8. A 15% decrease means b = 0.85.
- Mixing up growth and decayIf the quantity goes down, b is between 0 and 1. “Grows 100%” means doubling (b = 2), not b = 1.
- Forgetting the length of one periodDoubles every 3 years gives the exponent t/3, not 3t. Test with t = 3: you should get exactly one doubling.
- Calling every increasing table exponentialCompare differences and ratios. Equal differences mean linear; equal ratios mean exponential.
- Wrong order of operationsIn 5·2³ do the power first: 5 × 8 = 40. (5·2)³ = 1,000 is a different expression.
- Rounding early or typing symbolsKeep the full decimal until the last step. In a typed answer write 1240, not $1,240.
Type the function, for example y = 400(3)^(x/2), and Desmos draws the curve. Click the y-axis crossing to read a. To find when a quantity reaches a goal, type the goal as a second line, such as y = 5000, and click the intersection point: its x-coordinate is the time. Without Desmos you would need logarithms, which the SAT does not ask for. Sliders for a and b show how each number changes the graph, and Desmos can also fit an exponential curve to a table (advanced).
- Type y = 400(3)^(x/2)
- Type y = 5000
- Click the crossing point: it is about (4.6, 5000), so it takes about 4.6 hours
- Type y = a(b)^x and accept the sliders to match a given graph
Use Desmos for “when does it reach…?” questions and for ugly numbers. For tables, simply divide each output by the one before it: that is faster than graphing.
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