Exponential functions and models — Digital SAT Math practice
Growth, decay, percent change over time.
Key terms
- Initial Value: The starting amount or y-intercept of the function when x equals zero.
- Growth Factor: The constant multiplier b that represents how much the quantity changes each period.
- Decay Factor: A value between zero and one that shows the fraction remaining after each time step.
- Relative Rate: The decimal form of the percent change used to calculate the growth or decay factor.
- Time Scaling: Adjusting the exponent by dividing the elapsed time by the period length for the rate.
Common traps
- Adding the percent directly to the initial value instead of multiplying by the growth factor each period.
- Forgetting to convert the percentage to a decimal before adding or subtracting one to find b.
- Using the wrong exponent when the rate applies to a different time interval than the question asks.
Key ideas to know
Exponential Model Structure
In the exponential model y = a × bx, what do the constants a and b represent?
a is the starting value at x = 0. b is the growth or decay factor per unit step of x.
Watch out: Confusing b with the percent change itself instead of converting r% to 1 + r/100 or 1 − r/100.
Converting Percent to Factor
Convert a 35% annual growth rate and a 12% annual decay rate into their respective exponential factors b.
Growth factor b = 1 + 0.35 = 1.35. Decay factor b = 1 − 0.12 = 0.88. Always convert the percent to a decimal first.
Watch out: Forgetting to subtract the decay percent from 1, or writing 2 instead of 1.35 for growth.
Repeated Percent Change Error
A stock drops 10% in January and another 15% in February. What is the total percent change? The student says it is a 25% drop because 10 + 15 = 25. Is this correct?
No. Percent changes multiply, they do not add. The total factor is (0.90)(0.85) = 0.765, meaning a 23.5% drop, not 25%.
Watch out: Adding or subtracting percent rates directly instead of multiplying their corresponding factors for each period.
Interpreting Fractional Exponents
A population grows by a factor of 3 every 6 years. Which equation correctly models this if P0 is the initial population? A) P = P0 × 3t B) P = P0 × 3t/6 C) P = P0 × 6t/3 D) P = P0 + 3t
B. When the exponent is t/k, the quantity changes by factor b every k units of time. Here, b=3 and k=6.
Watch out: Placing the time unit in the base instead of the denominator of the exponent, or confusing the growth factor with the time period.
Exponential vs Linear Growth
Table A shows values increasing by 5 each step. Table B shows values multiplying by 2 each step. Which table represents an exponential model?
Table B represents the exponential model because exponential functions grow by equal ratios, while linear models grow by equal differences.
Watch out: Assuming any rapidly increasing sequence is exponential without checking if the ratio between consecutive terms is constant.
Sample questions
Question 1
A town's population is currently 5,000 and grows by 3% each year. Which function models the population P after t years?
- P(t) = 5000(0.03)ᵗ
- P(t) = 5000(1.03)ᵗ
- P(t) = 5000 + 1.03t
- P(t) = 5000(3)ᵗ
Show answer and explanation
Answer: P(t) = 5000(1.03)ᵗ
Growth of 3% per year means multiplying by 1.03 each year: P(t) = 5000(1.03)ᵗ.
Question 2
A bacterial culture starts at 100 cells and doubles every 3 hours. How many cells are there after 9 hours?
- 600
- 300
- 800
- 900
Show answer and explanation
Answer: 800
9 hours is 3 doublings: 100 x 2^3 = 800.
Question 3
If 2^(x + 1) = 32, what is the value of x?
- 4
- 5
- 6
- 16
Show answer and explanation
Answer: 4
32 = 2^5, so x + 1 = 5 and x = 4.