Graphing Exponential Functions

Key Concepts

• Exponential Functions: Functions of the form ( a^x ) where ( a ) is a constant.
• Horizontal Asymptote: The line ( y = \text{constant} ) that the graph approaches but never touches.
• Domain: All real numbers for exponential functions.
• Range: Depends on the horizontal asymptote; typically ((0, \infty)) or ((a, \infty)) where ( a ) is the horizontal asymptote.

Example 1: Basic Exponential Function

• Function: ( f(x) = 2^x )
• Points:
  • ( (0, 1) ) (\rightarrow) y-intercept
  • ( (1, 2) )
• Graph:
  • Starts from the horizontal asymptote at the x-axis (( y = 0 )) and increases.
• Domain: All real numbers
• Range: ( (0, \infty) )

Example 2: Fractional Base

• Function: ( f(x) = \left(\frac{1}{3}\right)^x )
• Change to: ( f(x) = 3^{-x} )
• Points:
  • ( (0, 1) ) (\rightarrow) y-intercept
  • ( (-1, 3) )
• Graph:
  • Starts from the horizontal asymptote at ( y = 0 ) and increases towards the plotted points.
• Domain: All real numbers
• Range: ( (0, \infty) )

Example 3: Shifted Exponential Function

• Function: ( f(x) = 3^{x-2} + 1 )
• Horizontal Asymptote: ( y = 1 )
• Points:
  • ( (2, 2) )
  • ( (3, 4) )
• Graph:
  • Starts from the horizontal asymptote (( y = 1 )) and increases.
• Domain: All real numbers
• Range: ( (1, \infty) )

Example 4: Decreasing Exponential Function

• Function: ( f(x) = 5 - 2^{3-x} )
• Horizontal Asymptote: ( y = 5 )
• Points:
  • ( (2, 3) )
  • ( (3, 4) )
• Graph:
  • Starts from the horizontal asymptote (( y = 5 )) and decreases.
• Domain: All real numbers
• Range: ( (-\infty, 5) )

Graph Reflections and Quadrants

• 2^x: Increases in Quadrant I.
• 2^{-x}: Reflects over the y-axis, increases in Quadrant II.
• -2^x: Reflects over the x-axis, increases in Quadrant IV.
• -2^{-x}: Reflects across the origin, increases in Quadrant III.

Summary

• Use the structure and transformations of the function to determine key points and asymptotes.
• Domain is always all real numbers.
• Range is determined by the horizontal asymptote.
• Consider reflections and shifts in determining the graph's path through quadrants.