Lecture Notes: Graphing Exponential Functions

Introduction to Exponential Functions

• Focus on graphing exponential functions.
• Specific focus on functions with base 2 and later base e.

Basic Function Example

• Function: ( y = 2^x )
  • Graphing Steps:
    1. Create a table of values for x.
    2. Choose x-values: 0 and 1.
    3. Calculate corresponding y-values:
       • ( 2^0 = 1 )
       • ( 2^1 = 2 )
  • Plot Points: (0, 1) and (1, 2).
  • Horizontal Asymptote: y = 0 (x-axis).
  • Domain: ( (-\infty, \infty) )
  • Range: ( (0, \infty) )

Example with Horizontal Shift

• Function: ( y = 3^{x+1} - 2 )
  • Shift:
    • Down 2 units ((-2)).
    • Left 1 unit ((x+1)).
  • New x-values after shift: -1 and 0.
  • Calculate y-values:
    • For ( x = -1: y = 1 )
    • For ( x = 0: y = 1 )
  • Horizontal Asymptote: y = -2.
  • Domain: ( (-\infty, \infty) )
  • Range: ( (-2, \infty) )

Base e Function Example

• Function: ( y = e^x - 1 )
  • Base e Approximation: ( e \approx 2.7 )
  • Choose x-values: 0 and 1.
  • Calculate y-values:
    • ( x = 0: y = 0 )
    • ( x = 1: y \approx 1.7 )
  • Horizontal Asymptote: y = -1.
  • Domain: ( (-\infty, \infty) )
  • Range: ( (-1, \infty) )

Function with Reflection

• Function: ( y = 3 - e^{x-2} )
  • Horizontal Shift: Right 2 units.
  • New x-values: 2 and 3.
  • Horizontal Asymptote: y = 3.
  • Reflection: Due to the negative sign.
  • Calculate y-values:
    • ( x = 2: y = 2 )
    • ( x = 3: y \approx 0.3 )
  • Domain: ( (-\infty, \infty) )
  • Range: ( (-\infty, 3) )

Summary

• The domain for exponential functions is generally ( (-\infty, \infty) ).
• The range depends on the horizontal asymptote and any transformations applied to the function.
• Understanding horizontal shifts and reflections is key to correctly graphing exponential functions.