Exponential Growth
Lesson notes
Overview
Quantities that multiply by a constant factor each step: money, bacteria, viral spread. This topic is rated beginner and is number 11 of 245 in Mathematics. Core ideas and first intuitions. No prior knowledge needed.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
In exponential growth a quantity is multiplied by the same factor in every equal time step. At first the increase seems slow, then it accelerates until it becomes enormous, because each new amount adds to the base that grows next. The base b above 1 sets the speed, and a is the starting value. Compound interest, a spreading virus, and bacteria dividing are all examples. A helpful check is the doubling time, which is constant however large the quantity has become.
Reading the relationship
The expression "y = a·bˣ" is a compact summary of the idea. It uses the symbols y, a, b; name what each one stands for before you start. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Start by only moving one slider at a time. Say aloud what you expect, then check.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Square Root Functions”, the topic before it in Mathematics. Once you are comfortable, continue with “Exponential Decay”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Exponential Growth” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Exponential Growth?
Quantities that multiply by a constant factor each step: money, bacteria, viral spread. It is a beginner topic in Mathematics.
How do I use the Exponential Growth simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression y = a·bˣ packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Core ideas and first intuitions. No prior knowledge needed. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Square Root Functions”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.
Quick quiz
More in Mathematics
See all →Exponential Decay
The opposite of growth: radioactive material, drug concentration and cooling follow e^(−kt).
Logarithms
A logarithm answers “what power gives this number?” It is the inverse of an exponential.
Euler’s Number e
The base for which the slope of eˣ equals its height, making it the natural language of growth.