Exponential Functions: The Basics

Linear growth adds the same amount each step: $10, 20, 30, 40$. Exponential growth multiplies by the same number each step: $10, 20, 40, 80$.

Every exponential function looks like $y = a \cdot b^{x}$. The number $a$ is the starting amount, the value when $x$ is $0$. The base $b$ is the growth factor, the number you multiply by each step. The exponent $x$ is how many steps you take.

The one move is turning a percent into $b$. Growing by 7 percent gives $b = 1 + 0.07 = 1.07$. Shrinking by 7 percent gives $b = 1 - 0.07 = 0.93$. When $b$ is bigger than $1$ the amount grows. When $b$ is between $0$ and $1$ the amount shrinks, which is called decay. The $y$-intercept is always the point $(0, a)$.