You're choosing between two job offers. Both start at £20,000 a year. Offer A promises a raise of £2,000 every year. Offer B promises a raise of 5% every year. Which deal is better?

For a long while, it's Offer A. After ten years A pays £40,000 while B pays only about £32,600 — adding £2,000 beats multiplying by 1.05, year after year. But the two rules grow in completely different ways. Offer A grows by adding the same amount each year, so its graph is a straight line. Offer B grows by multiplying by the same factor each year — and each multiplication acts on an ever-bigger salary, so the raises themselves keep swelling. Around year 27 the curve overtakes the line, and it never looks back: by year 40, Offer B pays about £140,800 against A's £100,000, and the gap is exploding.

(Here x is years worked and y is the salary in £1000s.) That is the whole story of this page: quantities that grow by adding give straight-line (linear) graphs, quantities that grow by multiplying give exponential graphs — and multiplication always wins eventually, spectacularly.

The exponential rule

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) — the factor y gets multiplied by every time x goes up by 1 — and a sets the starting height. In the job-offer story, Offer B is exactly this rule with a = 20{,}000 and b = 1.05. 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.

Worked example: plotting y = 2^{x} from a table

The cleanest exponential of all has a = 1 and b = 2. Build a table exactly as you would for any graph — remember that a negative power means a fraction (2^{-1} = \tfrac{1}{2}) and anything to the power 0 is 1:

Read along the row and you can see the exponential's fingerprint: every time x steps up by 1, y is multiplied by the same factor — here \times 2, the doubling signature. Equal steps in x, equal multipliers in y. A straight line has equal steps and equal additions; that one word — additions versus multipliers — is the whole difference. Plot the points and join them with a smooth curve: hugging the axis on the left, sweeping steeply up on the right, passing through (0,\ 1).

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. Compound interest, a spreading virus, a doubling population.
• If 0 < b < 1 it shows decay — it falls toward the axis, flattening out. A cooling drink, a depreciating car, a fading dose of medicine.

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. Halving something forever makes it tiny, but never makes it nothing.

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. Notice three things as you slide: the curve always pins itself to (0,\ 1); at exactly b = 1 it collapses to a flat line (multiplying by 1 changes nothing); and a decay curve like b = 0.5 is the mirror image of the growth curve b = 2, flipped left-to-right.

Worked example: a car losing 20% a year

A car is bought for £15,000 and loses 20% of its value every year. Losing 20% means keeping 80%, so each year the value is multiplied by 0.8 — that single move, turning a percentage change into a multiplier, is the key to every question like this. After n years:

Track it year by year:

• After 1 year: 15000 \times 0.8 = £12{,}000
• After 2 years: 12000 \times 0.8 = £9{,}600
• After 3 years: 9600 \times 0.8 = £7{,}680 — or in one go, 15000 \times 0.8^{3} = 15000 \times 0.512 = £7{,}680.

Notice the losses shrink: £3,000 in the first year, £2,400 in the second, £1,920 in the third — 20% of an ever-smaller value. That's why a decay graph starts steep and flattens out, gliding down toward the axis:

Linear or exponential? Interrogate the table

Exam questions love to hand you a bare table and ask what kind of rule made it. The test is two quick checks: look at the differences (what's added each step) and the ratios (what's multiplied each step). Same difference every step → linear. Same ratio every step → exponential. Try these two:

Both start 5 \to 10, so the first step tells you nothing! But A keeps adding +5 (constant difference: linear, y = 5x + 5), while B keeps multiplying \times 2 (constant ratio: exponential, y = 5 \cdot 2^{x}). By x = 10, rule A reaches 55 and rule B reaches 5120. Always check more than one step, and check ratios as well as differences.

Reading real-life graphs

Many real situations are read straight from the shape of a graph, no formula needed. The skill is to match each piece of the picture to a piece of the story.

On a distance–time graph of a journey: a straight rising line is steady speed, a steeper line is faster, and a flat stretch means stopped — time passes but distance doesn't change. So "cycled to a friend's house, stayed an hour, rode home faster" reads as: a rising line, a long horizontal shelf, then a steeper line coming back down to zero distance.

Or picture water pouring at a steady rate into a container, plotting depth against time. In a straight-sided glass the depth rises in a straight line. In a cone-shaped (ice-cream-cone) glass, the narrow bottom fills fast and the wide top fills slowly — so the depth graph starts steep and flattens. Read it backwards, too: a depth graph that starts gentle and then shoots up means the container is wide at the bottom and narrow at the top, like a conical flask. The graph is a portrait of the container.

And growth and decay graphs — a population, savings under compound interest, a cooling cup of coffee — carry the exponential shapes you now know: curving ever upward for growth, sliding down toward (but never onto) a floor for decay.