Growth, decay and horizontal asymptotes

In \(y=ar^x+b\), the positive base r controls growth or decay. With \(a>0\), a base greater than 1 gives growth, while \(0<r<1\) gives decay. The y-intercept is \(a+b\), because \(r^0=1\). For \(r\ne1\), the horizontal asymptote is \(y=b\).

Example

For \(y=3(1/2)^x+2\), the graph decreases, crosses the y-axis at 5 and approaches 2 from above as x increases.

A negative a reflects the graph vertically relative to its asymptote. If \(r=1\), the expression is the constant \(a+b\), not a growth or decay curve. An asymptote need not be the x-axis.