Key Concepts

• The graph of the function [latex]f\left(x\right)={b}^{x}\\[/latex] has a y-intercept at [latex]\left(0, 1\right)\\[/latex], domain [latex]\left(-\infty , \infty \right)\\[/latex], range [latex]\left(0, \infty \right)\\[/latex], and horizontal asymptote [latex]y=0\\[/latex].
• If [latex]b>1\\[/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0\\[/latex], 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 [latex]y=0\\[/latex].
• The equation [latex]f\left(x\right)={b}^{x}+d\\[/latex] represents a vertical shift of the parent function [latex]f\left(x\right)={b}^{x}\\[/latex].
• The equation [latex]f\left(x\right)={b}^{x+c}\\[/latex] represents a horizontal shift of the parent function [latex]f\left(x\right)={b}^{x}\\[/latex].
• Approximate solutions of the equation [latex]f\left(x\right)={b}^{x+c}+d\\[/latex] can be found using a graphing calculator.
• The equation [latex]f\left(x\right)=a{b}^{x}\\[/latex], where [latex]a>0\\[/latex], represents a vertical stretch if [latex]|a|>1[/latex] or compression if [latex]0<|a|<1\\[/latex] of the parent function [latex]f\left(x\right)={b}^{x}\\[/latex].
• When the parent function [latex]f\left(x\right)={b}^{x}\\[/latex] is multiplied by –1, the result, [latex]f\left(x\right)=-{b}^{x}\\[/latex], is a reflection about the x-axis. When the input is multiplied by –1, the result, [latex]f\left(x\right)={b}^{-x}\\[/latex], is a reflection about the y-axis.
• All translations of the exponential function can be summarized by the general equation [latex]f\left(x\right)=a{b}^{x+c}+d\\[/latex].
• Using the general equation [latex]f\left(x\right)=a{b}^{x+c}+d\\[/latex], we can write the equation of a function given its description.