Unit 2: Exponential & Logarithmic Functions

Exponential Data Modeling

Fitting an exponential function to context and data.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Geometric Sequences

Building an Exponential Model from What You're Given

An exponential model needs exactly two numbers: a starting value and a constant growth factor. Before reading on, think about how you'd find both if you were only told the starting value and a growth percentage — versus only being handed two data points, with no percentage stated at all.

Definition — Exponential Model

An exponential model has the form f(x) = a · bˣ, where a = f(0) is the initial value and b is the constant growth (b > 1) or decay (0 < b < 1) factor per unit of x. When growth is described as a percentage rate r, b = 1 + r for growth or b = 1 − r for decay.

Worked Example — Modeling from a Growth Rate

A population starts at 200 and grows 8% per year. Since b = 1 + 0.08 = 1.08, the model is P(t) = 200(1.08)ᵗ. Find P(5): 1.08⁵ = 1.4693280768 (computed by repeated squaring: 1.08² = 1.1664, 1.08³ = 1.259712, 1.08⁴ = 1.36048896, 1.08⁵ = 1.4693280768), so P(5) = 200 × 1.4693280768 ≈ 293.9. After 5 years, the population has grown from 200 to about 294.

Worked Example — Modeling from Two Data Points

Find an exponential model given f(0) = 10 and f(3) = 80, with no growth rate stated. Since a = f(0) = 10, substitute the second point: 10 · b³ = 80, so b³ = 8, giving b = 2. The model is f(x) = 10 · 2ˣ. Check: f(3) = 10 · 2³ = 10 · 8 = 80, matching the given value. Here b = 2 means the quantity doubles every 1 unit of x.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

When two data points are given instead of a rate, divide the two equations f(x₁) = a·bˣ¹ and f(x₂) = a·bˣ² to eliminate a first — it isolates b as a single power you can solve directly, the way b³ = 8 did above.

Common Mistakes

Writing b = 0.08 directly from an '8% growth' rate, instead of 1.08.
A growth rate describes the change, not the new total — the base itself has to include the 100% that's still there plus the 8% added: b = 1 + 0.08 = 1.08.
Assuming a = f(0) can be read off directly even when the given data doesn't include x = 0.
If neither given point has x = 0, solve for a and b together as a system, the same way b was isolated above, rather than guessing a from a point that isn't actually the initial value.

Key Takeaways

An exponential model f(x) = a · bˣ needs an initial value a and a growth/decay factor b.
A percentage growth rate r converts to a base as b = 1 + r (growth) or b = 1 − r (decay).
Given two data points instead of a rate, dividing one function equation by the other isolates b as a solvable power.

Summary

Fitting a model is only half the picture — the next lesson studies the function that exactly undoes an exponential, and uses it to straighten exponential data into a line.

Track your progress

← Geometric Sequences Back to Unit 2: Exponential & Logarithmic Functions Logarithms as Inverses→