3.1 | Exponential Functions and Their Graphs


The exponential function \(f\) with base \(a\) is denoted by
$$
f(x) = a^x
$$
where \(a>0, a\not = 1,\) and \(x\) is any real number.
In general, to graph
$$
f(x) = a \cdot b^{kx-c} + d
$$
you can rely on three defining characteristics of the graph of \(f\):

  • The horizontal asymptote: \(y = d\)
  • The \(y\)-intercept (one always exists)
  • An additional point

Plot the horizontal asymptote using a dashed line, plot the \(y\)-intercept and an additional point. Use a smooth curve to connect the two points and extend the graph so that the curve becomes asymptotic to the line \(y = d\). You should also extend the curve in the other direction and so that the curve moves away from the line \(y = d\). Essentially you want to preserve the general shape of the basic exponential function, when graphing its variations.

After \(t\) years, the balance \(A\) in an account with principal \(P\) and annual interest rate \(r\) (in decimal form) is given by the following formulas.

  1. For \(n\) compoundings per year: \(\displaystyle A = P \left(1 + \frac{r}{n} \right)^{nt}\)
  2. For continuous compounding: \(A = Pe^{rt}\)

Graph
$$
f(x) = -2^{x-1} +3.
$$


E 3.1 Exercises