Exponential Functions and Models

Growth, Decay, and the Power of e — A TLDR Primer

Exponential functions show up on every algebra 2 and precalculus exam — and most students hit a wall the moment the variable moves into the exponent. If you have a test coming up, a homework problem about radioactive decay you can't crack, or a parent trying to walk a kid through compound interest, this guide is built for exactly that moment.

TLDR: Exponential Functions and Models covers everything from the ground up, short by design. You'll learn what makes a function exponential (and why it's not the same as a power function), how to read and sketch graphs, and why the number e keeps appearing in science and finance. The guide builds real-world models for population growth, compound interest, radioactive half-life, and Newton's law of cooling — then shows you, step by step, how to solve exponential equations using logarithms to answer questions like when does the population hit one million? A final section connects these ideas to biology, epidemiology, and finance, and points toward calculus for readers who want to go further.

This is an exponential growth and decay math primer written for high school students in grades 9–12 and early college students who need a clear, fast orientation — not a textbook. Every term is defined, every concept is anchored to a worked example, and nothing is padded.

If algebra 2 exponential functions have felt slippery, pick this up and read it once before your next class or exam.

What you'll learn

Recognize exponential functions and distinguish them from polynomial and linear functions
Graph exponential functions and identify key features: y-intercept, asymptote, growth vs. decay
Use the natural base e and convert between forms a*b^t and a*e^(kt)
Build exponential models for population growth, compound interest, and radioactive decay
Solve exponential equations using logarithms
Avoid common student errors involving rates, doubling time, and half-life

What's inside

1. What Is an Exponential Function?

Defines exponential functions, contrasts them with linear and power functions, and introduces the base, growth factor, and y-intercept.
2. Graphs and Behavior

Shows how to graph exponential functions, identify horizontal asymptotes, and describe end behavior for growth and decay.
3. The Number e and Continuous Growth

Introduces e as the natural base, derives it from compound interest, and connects a*b^t with a*e^(kt).
4. Modeling Growth and Decay

Builds exponential models for population, compound interest, radioactive decay, and Newton's law of cooling, with emphasis on doubling time and half-life.
5. Solving Exponential Equations

Uses logarithms to solve exponential equations and answer model questions like 'when will the population reach X?'
6. Why It Matters and What Comes Next

Connects exponential models to finance, biology, epidemiology, and the limits of exponential thinking, and previews logarithms and differential equations.

Published by Solid State Press · June 2026

TLDR STUDY GUIDES

Exponential Functions and Models

Growth, Decay, and the Power of e — A TLDR Primer

Solid State Press

Exponential Functions and Models

Growth, Decay, and the Power of e — A TLDR Primer

Published by Solid State Press.

Part of the TLDR Study Guides series.

1 What Is an Exponential Function?
2 Graphs and Behavior
3 The Number e and Continuous Growth
4 Modeling Growth and Decay
5 Solving Exponential Equations
6 Why It Matters and What Comes Next

Chapter 1

What Is an Exponential Function?

The single most important fact about an exponential function is where the variable sits: in the exponent, not the base.

An exponential function has the form

$f (t) = a \cdot b^{t}$

where $a$ is a nonzero constant, $b$ is a positive constant not equal to 1, and $t$ is the variable. The number $b$ is called the base, and $a$ is called the initial value (because when $t = 0$ , $f (0) = a \cdot b^{0} = a$ ).

How exponential differs from what you already know

You have seen two other families of functions. A linear function looks like $f (t) = m t + c$ — the variable $t$ is multiplied by a constant. A power function looks like $f (t) = t^{n}$ — the variable is raised to a fixed exponent. In an exponential function, the roles are flipped: the base is fixed and the exponent varies.

This distinction matters more than it looks. Linear functions grow by a constant amount each step. Exponential functions grow by a constant ratio each step. That one difference produces dramatically different behavior as $t$ gets large.

Compare these three tables, each starting at 2 when $t = 0$ :

$t$	Linear: $2 + 3 t$	Power: $2 t^{3}$	Exponential: $2 \cdot 3^{t}$
0	2	0	2
1	5	2	6
2	8	16	18
3	11	54	54
4	14	128	162
5	17	250	486

The linear function adds 3 every time. The power function grows faster than linear, but notice that at $t = 3$ it briefly matches the exponential — then the exponential leaves it behind entirely. By $t = 5$ , the exponential is nearly twice as large as the power function. This eventual dominance of exponential over polynomial growth is one of the defining features of the family.

The base determines growth or decay

About This Book

If you're a high school student who needs an exponential functions study guide for Algebra 2, Precalculus, or an AP math course — or a college freshman who got lost the first week of Calculus — this book is for you. It's also useful for tutors running a quick review session or parents trying to help a student prep for a test.

This exponential growth and decay math primer covers the core ideas from the ground up: what makes a function exponential, how to read and sketch graphs, and why the number e matters for students moving into continuous-growth problems. You'll work through compound interest and exponential equations, learn to set up half-life and doubling time models, and build the Algebra 2 exponential functions review skills that carry straight into Precalculus exponential models and beyond. A concise overview with no filler.

Read straight through once, then work every example yourself before checking the solution. Finish with the problem set at the end to confirm you've got it.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters — readable in one sitting.

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