Lecture Notes: Understanding Exponentials

Introduction

• Exploration of exponentials in-depth beyond basic graphing.
• Focus on why exponentials look the way they do.
• Importance of understanding exponentials for logarithms and inverse graphs.

What is an Exponential Function?

• Exponential Function: Takes a constant base (positive, not 1, not 0) and raises it to a variable exponent.
• Different from polynomials which raise a variable to a constant.
• Exponential functions are vital for modeling population growth and other phenomena.

Characteristics of Exponential Functions

• Base (a): Must be a positive constant; cannot be 0 or 1.
  • If the base is negative, the graph becomes non-continuous and disjointed.
  • Base of 1 results in a constant horizontal line (not useful).
• Behavior:
  • Base > 1: Graph is increasing.
  • 0 < Base < 1: Graph is decreasing.

Graphing Exponentials

• Key Points:
  • Evaluate at x = 0 gives y = 1 (y-intercept).
  • Evaluate at x = 1 gives y = base.
• Horizontal Asymptote: At y = 0.
• Domain: All real numbers.
• Range: Positive real numbers (never reaches 0 or becomes negative).

Effects of Base Value

• Numbers greater than 1 raised to positive exponents grow.
• Numbers between 0 and 1 raised to positive exponents shrink.
• Behavior of negative exponents creates reciprocals, not negative numbers.

Detailed Example

• Example of base 2:
  • Values like 2^x create increasing graphs.
  • Negative x-values create fractions, e.g., 2^-1 = 1/2.
• Example of base 1/2:
  • Values like (1/2)^x create decreasing graphs.
  • Positive x-values create smaller fractions.

Special Case: Euler’s Number (e)

• Euler Number (e): Approximately 2.7, transcendental number.
• Graph of e^x:
  • Increasing like base > 1 example (2^x).
  • Used extensively in natural sciences and mathematics.

Implications for Logarithms

• Exponential Functions are one-to-one:
  • Passes the horizontal line test.
  • Have inverses, crucial for understanding logarithms.

Conclusion and Next Steps

• Next: Practice graphing with transformations.
• Understanding exponentials is foundational for logarithms and further mathematical exploration.