Key Concepts

• The graph of the function $f\left(x\right)={b}^{x}$ has a y-intercept at $\left(0, 1\right)$, domain $\left(-\infty , \infty \right)$, range $\left(0, \infty \right)$, and horizontal asymptote $y=0$.
• If $b>1$, the function is increasing. The left tail of the graph will approach the asymptote $y=0$, and the right tail will increase without bound.
• If 0 < b < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $y=0$.
• The equation $f\left(x\right)={b}^{x}+d$ represents a vertical shift of the parent function $f\left(x\right)={b}^{x}$.
• The equation $f\left(x\right)={b}^{x+c}$ represents a horizontal shift of the parent function $f\left(x\right)={b}^{x}$.
• Approximate solutions of the equation $f\left(x\right)={b}^{x+c}+d$ can be found using a graphing calculator.
• The equation $f\left(x\right)=a{b}^{x}$, where $a>0$, represents a vertical stretch if $|a|>1$ or compression if $0<|a|<1$ of the parent function $f\left(x\right)={b}^{x}$.
• When the parent function $f\left(x\right)={b}^{x}$ is multiplied by –1, the result, $f\left(x\right)=-{b}^{x}$, is a reflection about the x-axis. When the input is multiplied by –1, the result, $f\left(x\right)={b}^{-x}$, is a reflection about the y-axis.
• All translations of the exponential function can be summarized by the general equation $f\left(x\right)=a{b}^{x+c}+d$.
• Using the general equation $f\left(x\right)=a{b}^{x+c}+d$, we can write the equation of a function given its description.