Summary: Graphs of Exponential Functions

Key Equations

General Form for the Translation of the Parent Function  f(x)=bx f(x)=bx f(x)=abx+c+df(x)=abx+c+d

Key Concepts

  • The graph of the function f(x)=bxf(x)=bx has a y-intercept at (0,1)(0,1), domain (,)(,), range (0,)(0,), and horizontal asymptote y=0y=0.
  • If b>1b>1, the function is increasing. The left tail of the graph will approach the asymptote y=0y=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=0y=0.
  • The equation f(x)=bx+df(x)=bx+d represents a vertical shift of the parent function f(x)=bxf(x)=bx.
  • The equation f(x)=bx+cf(x)=bx+c represents a horizontal shift of the parent function f(x)=bxf(x)=bx.
  • Approximate solutions of the equation f(x)=bx+c+df(x)=bx+c+d can be found using a graphing calculator.
  • The equation f(x)=abxf(x)=abx, where a>0a>0, represents a vertical stretch if |a|>1|a|>1 or compression if 0<|a|<10<|a|<1 of the parent function f(x)=bxf(x)=bx.
  • When the parent function f(x)=bxf(x)=bx is multiplied by –1, the result, f(x)=bxf(x)=bx, is a reflection about the x-axis. When the input is multiplied by –1, the result, f(x)=bxf(x)=bx, is a reflection about the y-axis.
  • All translations of the exponential function can be summarized by the general equation f(x)=abx+c+df(x)=abx+c+d.
  • Using the general equation f(x)=abx+c+df(x)=abx+c+d, we can write the equation of a function given its description.