AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-5)
Unit 2 · Topic 2.5
2.5 Exponential Function Context and Data Modeling
Exponential models describe quantities that change by the same percent over equal time steps: money earning interest, a growing population, a medicine leaving the body. You'll build them from a percent rate, a half-life, two data points or a regression, and read the base as a growth factor.
Key terms
- growth factor
- percent rate of change
- half-life
- natural base e
- exponential regression
When an exponential model fits
Use an exponential model when the outputs over equal-length input intervals are proportional, meaning each one is the same multiple of the one before. For whole-number inputs, that's repeated multiplication of a starting value by a constant.
Sometimes the data only become proportional after you subtract a constant. A cooling cup of coffee approaches room temperature, not 0, so the temperature minus room temperature is what decays exponentially.
Growth factor and percent change
In f(t) = a · bᵗ, the base b is the growth factor for each 1-unit increase in t.
A percent increase of r (as a decimal) gives b = 1 + r. Growing 3.5% per year means b = 1.035.
A percent decrease of r gives b = 1 − r. Losing 12% per hour means b = 0.88.
Half-life h: A(t) = a · (1/2)^(t/h), because the amount halves every h units. Doubling time d: A(t) = a · 2^(t/d).
Equivalent forms change the time unit
The same function can be written with different time units. If f(d) = 2ᵈ with d in days, then f(d) = (2⁷)^(d/7) = 128^(d/7): the quantity multiplies by 128 every week.
Going the other way, a yearly factor of 1.035 is a monthly factor of 1.035^(1/12) ≈ 1.002871, about 0.287% per month. You divide the exponent, not the percent: 3.5%/12 is close but not equal.
Building a model from information
From an initial value and a factor: write a · bᵗ directly.
From two points: write a system. If f(1) = 150 and f(4) = 1200, then a · b = 150 and a · b⁴ = 1200. Dividing gives b³ = 8, so b = 2 and a = 75.
From a context with a shift: apply transformations, such as T(t) = 70 + 130(0.9)ᵗ for coffee that starts at 200° in a 70° room.
From data: use exponential regression on a calculator, which returns a and b for y = a · bˣ.
The natural base e
The number e ≈ 2.718 shows up whenever growth happens continuously. Models are often written as a · e^(kt), where k > 0 means growth and k < 0 means decay. Since e^(kt) = (eᵏ)ᵗ, this is just an exponential function with base b = eᵏ.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Percent growth
A town has 1,200 people and grows 3.5% per year. Write a model for the population t years from now, predict it after 10 years, and give the monthly growth factor.
Show the solutionHide the solution
- Step 1: Growth factor per year: b = 1 + 0.035 = 1.035. Initial value 1,200.
- Step 2: P(t) = 1200(1.035)ᵗ.
- Step 3: P(10) = 1200(1.035)¹⁰ ≈ 1692.719. Round to a whole number of people: about 1,693.
- Step 4: Monthly factor: 1.035^(1/12) ≈ 1.002871, so P can also be written 1200(1.002871)^(12t).
Answer: P(t) = 1200(1.035)ᵗ; about 1,693 people after 10 years; monthly growth factor ≈ 1.002871 (about 0.287% per month).
- Example 2Calculator allowed
Half-life
A patient receives 80 mg of a medicine whose half-life in the body is 6 hours. Write a model for the amount left after t hours and find the amount after 15 hours.
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- Step 1: Every 6 hours the amount is multiplied by 1/2, so A(t) = 80(1/2)^(t/6).
- Step 2: A(15) = 80(1/2)^(15/6) = 80(1/2)^2.5.
- Step 3: (1/2)^2.5 ≈ 0.176777, so A(15) ≈ 14.142.
Answer: A(t) = 80(1/2)^(t/6); about 14.142 mg after 15 hours.
- Example 3Calculator allowed
Exponential regression
Data: (0, 3.1), (1, 4.4), (2, 6.5), (3, 9.4), (4, 13.8), (5, 20.1). Use exponential regression to model the data, interpret the base, and predict y at x = 6.
Show the solutionHide the solution
- Step 1: The ratios of consecutive outputs are all about 1.4 to 1.5, so an exponential model is reasonable.
- Step 2: Exponential regression gives y ≈ 3.064(1.456)ˣ.
- Step 3: The base 1.456 means y grows by about 45.6% for each 1-unit increase in x.
- Step 4: Using the stored equation, y(6) ≈ 29.165.
Answer: y ≈ 3.064(1.456)ˣ; y grows about 45.6% per unit of x; predicted y(6) ≈ 29.165.
Common mistakes
- Using the percent as the base: 3.5% growth means b = 1.035, not 3.5 or 0.035.
- Writing a half-life model as a(1/2)ᵗ when the half-life isn't 1. The exponent must be t divided by the half-life.
- Converting a yearly percent to monthly by dividing by 12. The correct monthly factor is the 12th root of the yearly factor.
On the exam
- The non-periodic modeling free-response question often uses an exponential model: you may solve for its constants from given points and interpret the base as a percent change.
- Round final answers to three decimal places, but keep full precision in between, and state units.
Connected topics
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Check yourself
4 questions on 2.5 Exponential Function Context and Data Modeling. Pick an answer to see if you got it, and why.
A medication has a half-life of 6 hours in the body. A patient has 80 milligrams of the medication in their body at time t = 0. Assuming exponential decay, how many milligrams remain after 15 hours?
The number of users of an app is modeled by U(d) = 50,000(1.02)ᵈ, where d is the number of days since launch. Which of the following best describes the change in the number of users each week, according to the model?
A bacteria culture has 320 cells at t = 1 hour and 720 cells at t = 3 hours. If the number of cells grows exponentially, how many cells does the model predict at t = 6 hours?
A savings account pays 4.8% annual interest, compounded monthly. If $2000 is deposited and no other deposits or withdrawals are made, which of the following is closest to the balance after 5 years?
0 of 4 answered