Exponential Growth and Decay Graphs
Exponential functions change by a constant percentage rather than a constant amount. That difference is what makes them outrun every polynomial eventually, and why they describe compound interest, population growth and radioactive decay alike.
Open this graph in the calculator
What you are looking at
The teal curve is y = 2ˣ, an exponential growth curve: every time x increases by 1, y doubles. The rose curve is y = 0.5ˣ, exponential decay: every step right halves the value.
The two are mirror images across the y-axis, because 0.5ˣ equals 2⁻ˣ. Both pass through (0, 1), since anything raised to the power zero is 1.
The horizontal asymptote
Both curves approach the x-axis without ever touching it. Halving a positive number repeatedly makes it very small, but never zero. The line y = 0 is a horizontal asymptote.
This is why the range is (0, ∞) with a round bracket at zero: every output is strictly positive. It is also why an exponential model can never predict a value of exactly zero, which is a genuine limitation when modelling something that does reach zero.
Growth against polynomial growth
Exponential growth is slow at first and then overwhelming. Compare y = 2ˣ with y = x² and the parabola is ahead for small x, but by x = 10 the exponential is at 1024 against 100, and by x = 20 it is over a million against 400.
That crossover is the practical meaning of "exponential". It is also why an exponential trend extrapolated far beyond the data produces absurd predictions, and why real growth curves eventually flatten into an S shape as something runs out.
Where these turn up
Compound interest is exponential: money grows by a fixed percentage each period. Radioactive decay is exponential with a base below 1, and the half-life is the time for the quantity to halve. Populations grow exponentially while resources are unlimited, and cooling objects lose heat exponentially towards the ambient temperature.
| Situation | Form | What the base means |
|---|---|---|
| Compound interest | A(1 + r)ᵗ | Growth factor per period |
| Radioactive decay | N₀(1/2)^(t/h) | Halving every half-life h |
| Population growth | P₀eʳᵗ | Continuous growth at rate r |
| Cooling | T∞ + (T₀ − T∞)e^(−kt) | Approach to ambient temperature |
Questions
What makes e special as a base?
e ≈ 2.71828 is the base for which the curve y = eˣ has a slope exactly equal to its own height at every point. That property makes calculus with exponentials far simpler, which is why e appears throughout science even though any base could be used.
Is y = xˣ exponential?
No. An exponential has a constant base and a variable exponent. In xˣ both change, which makes it neither exponential nor polynomial, and it is only defined for positive x in the reals.
How do I find the equation from two points?
For y = abˣ, divide one equation by the other to eliminate a and solve for b, then substitute back. Or enter both points in the Data tab and run an exponential regression.