Topic 2.5: Exponential Function Context and Data Modeling — AP Precalculus

📌 Core Idea

Sometimes exponential data is vertically translated — the output values don't show a proportional pattern at first. By adding or subtracting a constant, we can reveal the hidden proportional growth and write the exponential equation.

🔍 Revealing Exponential Growth with Vertical Translations

The Key Idea

If you can add or subtract a constant k from the output values so that the adjusted outputs are proportional over equal-length input intervals, then the original function is exponential.

If f(x) − k shows proportional growth → f(x) = a · b^x + k

✏️ Example 1 — Finding the Vertical Translation

For each table, find the constant to add/subtract to reveal proportional growth, then write the equation.

Part a — f(x)

x	f(x)	f(x) − 1
0	7	6
1	13	12
2	25	24
3	49	48
4	97	96

Ratio: × 2 each step

f(x) = 6(2)^x + 1

Part b — g(x)

x	g(x)	g(x) + 2
0	2	4
1	10	12
2	34	36
3	106	108
4	322	324

Ratio: × 3 each step

g(x) = 4(3)^x − 2

Part c — h(x) (decay)

x	h(x)	h(x) + 1
0	63	64
1	31	32
2	15	16
3	7	8
4	3	4

Ratio: ÷ 2 each step

h(x) = 64(2)^−x − 1

📐 Modeling Exponential Functions from Data

Two Ways to Build a Model f(x) = ab^x

An exponential model can be constructed if given either:

1. The common ratio b and initial value a (a = f(0))
2. Two input-output pairs — set up two equations and solve for a and b

✏️ Example 2 — Two Input-Output Pairs

Fans in a stadium: F(t) = a(b)^t. There are 47 fans at t = 2 min and 2602 fans at t = 20 min.

F(2) = a(b)² = 47
F(20) = a(b)²⁰ = 2602

Divide the second equation by the first to eliminate a, then solve for b. Use a calculator's ExpReg (exponential regression) to verify.

✏️ Example 3 — Non-Integer Inputs

g(x) = ab^x with g(3) = 21.54 and g(8) = 3.62.

g(3) = ab³ = 21.54
g(8) = ab⁸ = 3.62

Solving: a ≈ 62.80, b ≈ 0.70

🌿 The Natural Base: e

e ≈ 2.718

The natural number e is often used as the base in exponential functions that model real-world contextual scenarios.

✏️ Example 4 — Working with e

Bacteria: B(t) = 17e^0.31t

(a) Rewrite as A(t) = a · b^t:
   A(0) = B(0) → a · b⁰ = 17e⁰ → a = 17

(b) AROC from t = 1 to t = 7:
   [B(7) − B(1)] / (7 − 1) ≈ 20.95

(c) When is B(t) = 1000?
   t ≈ 13.14 hours

📈 Percent Change Context

✏️ Example 5 — Deer Population

50 deer introduced; population grows 13% per year. D(t) = a · b^t

(a) D(0) = a = 50; b = 1.13 (13% growth)
D(t) = 50 · (1.13)^t

(b) After 6 years: D(6) = 50(1.13)⁶ ≈ 104.1 deer

(c) When does D(t) = 5000? t ≈ 37.68 years

(d) Rewrite in months: D(t) = 50 · (1.13)^t/12

Key: Growth rate r% → b = 1 + r/100. Decay rate r% → b = 1 − r/100.

Card 1 of 5

Concept

How do you identify if a vertically-translated function is exponential?

Tap to reveal answer ✨

Answer

Add/subtract a constant k so the adjusted outputs are proportional

If f(x) − k shows a constant ratio over equal intervals, then f(x) = a·bˣ + k. The constant k is the vertical translation.

Topic 2.5

Exponential Context & Modeling Quiz

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Question 1 of 4

The table shows values of f(x). After subtracting 1 from each output, the adjusted values are 6, 12, 24, 48, 96. What equation models f(x)?

Question 2 of 4

A deer population starts at 50 and grows 13% per year. Which function models the population after t years?

Question 3 of 4

B(t) = 17e^0.31t models bacteria. Which gives the initial value a if we rewrite as A(t) = a · b^t?

Question 4 of 4

g(x) = ab^x with g(3) = 21.54 and g(8) = 3.62. Which pair of equations can be used to find a and b?

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Keep going!

Exponential Function Context& Data Modeling

📌 Core Idea

The Key Idea

Two Ways to Build a Model f(x) = abx

✏️ Example 2 — Two Input-Output Pairs

✏️ Example 3 — Non-Integer Inputs

✏️ Example 4 — Working with e

✏️ Example 5 — Deer Population

💬 Ask a Question

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Exponential Function Context
& Data Modeling

Two Ways to Build a Model f(x) = ab^x