# 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)
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: $$A = Pe^{rt}$$
$$f(x) = -2^{x-1} +3.$$