Exponential Functions Notes

Exponential Functions Overview

• Definition: Exponential function = $A \cdot b^{x}$ where:

  • $A$ and $b$ are constants
  • A > 0, b > 0, and $b \neq 1$

• Domain: All real numbers (no values that yield an undefined result)

• Graphing:

  • Example: $y = 3^{x}$
  • Points:
    • $x = -1: ( -1, \frac{1}{3})$
    • $x = 0: (0, 1)$
    • $x = 1: (1, 3)$
    • $x = 2: (2, 9)$
    • Has a horizontal asymptote at $y = 0$ (approaches but never touches the x-axis)

• Reflection:

  • $y = \left(\frac{1}{3}\right)^{x}$ is a reflection of $y = 3^{x}$ across the y-axis

• Intercepts:

  • No x-intercepts for exponential functions (never crosses x-axis)
  • Y-intercept at $y = A$ when $x = 0$

• Transformations:

  • Horizontal asymptote remains at $y = 0$ unless vertical transformations are applied
  • Range: $]0, \infty[$

• Growth vs. Decay:

  • Exponential Growth: Base b > 1, graph increases from left to right
  • Exponential Decay: Base b < 1, graph decreases from left to right