An exponential graph comes from a rule where the variable sits in the power — built on the index laws:

Here b is the base (a positive number) and a sets the starting height. Unlike the cubic and reciprocal graphs, an exponential curve doesn't just rise quickly — the rate of rise itself keeps increasing, so it sweeps upward faster and faster.

Growth, decay, and the key features

The base b decides the whole shape:

• If b > 1 the graph shows growth — it climbs ever more steeply to the right.
• If 0 < b < 1 it shows decay — it falls toward the axis, flattening out.

Two features hold either way. Every exponential graph y = a \cdot b^{x} passes through (0,\ a), because b^{0} = 1. And it gets ever closer to the x-axis but never reaches zero — the axis is an asymptote the curve only approaches.

Try it live

Here is y = b^{x} (so the curve starts at (0,\ 1)). Slide the base b across 1 and watch the curve flip between growth and decay.

Reading real-life graphs

Many real situations are read straight from the shape of a graph, no formula needed. A distance–time graph tells a journey: a straight rising line is steady speed, a steeper line is faster, and a flat stretch means stopped. A growth graph — a population, savings with compound interest, or a virus spreading — often curves upward like an exponential, while a cooling cup of coffee curves downward like decay. The skill is to match the picture to the story it tells.