Concept preview

What you will be able to do

1. 01Model growth and decay with y = ab^x and read its parameters
2. 02Compute compound and continuous interest, doubling times, and half-lives
3. 03Solve exponential equations by matching bases and analyse transformations

An exponential function has the variable in the exponent: $y = a\,b^{x}$, where the base $b > 0$, $b \ne 1$. The constant $a$ is the initial value — the $y$-intercept, since $b^{0} = 1$. For a positive initial value $a>0$, if $b>1$ the quantity grows; if $0<b<1$ it decays. A negative $a$ reflects the graph and reverses its increasing/decreasing direction; $a=0$ gives the zero function. Unlike polynomials, whose growth is fixed by their degree, exponentials multiply by the same factor $b$ over each equal step, which is what makes them explode (or vanish) so quickly.