eˣ

Exponential Functions

Growth, decay, and the power of constant ratios.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualA curve that climbs by multiplying — not adding. Each step right multiplies the height by the same factor.

Algebra $y = b^x$ where $b>0$ , $b\neq 1$ .

y = b^x

VisualSlide b above 1 — the curve explodes upward. Slide b between 0 and 1 — it decays toward zero.

AlgebraIf $b>1$ : growth. If $0<b<1$ : decay. The horizontal asymptote is $y=0$ (or $y=c$ when shifted).

VisualP₀ scales the starting height, k controls the speed, c shifts the asymptote up or down.

Algebra $y = P_0 \cdot b^{kx} + c$ . Asymptote: $y=c$ . y-intercept: $P_0 + c$ .

y = P_0\, b^{kx} + c

VisualPopulation doubling, radioactive decay, compound interest — all follow this single curve.

AlgebraPopulation: $P(t)=P_0(1+r)^t$ . Half-life: $N(t)=N_0(\tfrac12)^{t/h}$ .

A = P\left(1+\frac{r}{n}\right)^{nt}

Exponential growth

y = ab^x,\; b>1

Multiplies by b each unit

Compound interest

A = P(1+\tfrac{r}{n})^{nt}

n times per year

Continuous growth

A = Pe^{rt}

Limit as n→∞

Mark this lesson complete to update your mastery.

easyexponentials

Evaluate

2^5

easyexponentials

Does

y=(\tfrac12)^x

represent growth or decay?

mediumexponentials

A population of 1000 grows at 8% per year. Find the population after 5 years (round to nearest whole).

mediumexponentials

Find the horizontal asymptote of

y=3(2)^x - 5

hardexponentials

A substance has half-life 8 days. If you start with 200 g, how much remains after 24 days?