Exponential Functions
Exponentials multiply by a constant factor each step: y = a·bˣ.
About this topic
An exponential function has the variable in the exponent, f(x) = a · bˣ with b greater than 0 and b ≠ 1. If b is greater than 1 it grows, and if b is between 0 and 1 it decays. Unlike linear functions, which add a constant, exponentials multiply by a constant factor each step.
The graph passes through (0, a), has a horizontal asymptote at y = 0 that it approaches but never reaches, and its domain is all reals. For f(x) = 3 · 2ˣ, the values at x = 0, 1, 2, 3 are 3, 6, 12, 24, doubling each time. For (1/2)ˣ, the values halve, producing a decay curve.
Exponentials describe populations, bacteria cultures, spread of information, radioactive decay, cooling coffee and compound interest. The famous rice-on-the-chessboard story shows how doubling goes from tiny to enormous.
A common mistake is to confuse x² with 2ˣ. In x² the variable is the base and the growth is polynomial, while in 2ˣ the variable is the exponent and the growth eventually beats any polynomial. Remember that the starting value a is the y-intercept and that b is the growth factor, not the percent rate.
Key ideas
- Domain is all reals, range is positive
- Passes through (0, a)
- b > 1 grows, 0 < b < 1 decays
Quick check
Use cases: where Exponential Functions helps
- Homework: review the key idea behind Exponential Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Exponential Functions without notes.
- Visual learning: play with the interactive lab to see how Exponential Functions changes when you adjust its inputs.
- Teaching a friend: explain Exponential Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Exponential Functions?
Exponentials multiply by a constant factor each step: y = a·bˣ.
What will I learn in Exponential Functions?
Domain is all reals, range is positive. Passes through (0, a). b > 1 grows, 0 < b < 1 decays.
Is Exponential Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Exponential Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Exponential Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.