Basic exponentials

An exponential function is written $y = a \cdot b^x$, where the variable $x$ appears in the exponent. Its defining feature is that increasing $x$ by $1$ multiplies $y$ by the constant factor $b$, instead of adding a fixed amount as a straight line would.

Whether the curve grows or decays depends on $b$. When $b > 1$ each step multiplies by more than one, so the values rise faster and faster: exponential growth. When $0 < b < 1$ each step multiplies by less than one, so the values shrink towards zero without ever reaching it: exponential decay. Both curves pass through $(0, a)$.

A useful consequence: because an exponential never turns around, on a domain that starts at $x = 0$ its largest or smallest value always sits at $x = 0$. A decreasing (decay) function takes its maximum at $x = 0$, while an increasing (growth) function takes its minimum at $x = 0$ — in each case that value is the starting amount $a$.

This multiplicative behaviour is why percentage change is exponential. A quantity decaying $8\%$ per hour is multiplied by $0.92$ each hour, not reduced by a flat amount, so its graph curves rather than falling in a straight line.

A $100$ g sample decays $8\%$ per hour, so each hour it is multiplied by $1 - 0.08 = 0.92$: after one hour $100 \times 0.92 = 92$ g, and after two hours $92 \times 0.92 \approx 85$ g. The drop is not a flat $8$ g each hour, which is what makes it exponential rather than linear.

For the growth function $y = 5 \cdot 2^x$, the starting value at $x = 0$ is $5$, and every increase of $1$ in $x$ doubles $y$: $5, 10, 20, 40, \dots$ So $y$ is multiplied by $2$ at each step rather than having a fixed amount added.

Consider the decay function $y = 100 \cdot (0.92)^x$ for $x \ge 0$. Its value is largest at the start: at $x = 0$, $y = 100$, and it only decreases afterwards, so its maximum is at $x = 0$. A growth function behaves the opposite way, taking its minimum at $x = 0$.