Understanding Exponential Functions and Growth

Mar 4, 2025

Chapter 7: Exponential and Logarithmic Functions

Section 7.1: Exponential Functions - Growth and Decay

Key Concepts

  • Exponential Functions: Differ from polynomial functions where the variable is the exponent rather than the base.
  • General Form: ( y = a \cdot b^x )
    • a: Initial amount (Y-intercept on the graph).
    • b: Growth factor (determines if the function represents growth or decay).

Growth and Decay

  • Growth: Occurs if ( b > 1 ).
  • Decay: Occurs if ( 0 < b < 1 ).

Graphing Exponential Functions

  • Create a table of values to plot the graph.
  • Evaluate the function at different points to obtain coordinates.
  • Y-intercept: Occurs at ( (0, a) ).
  • Reciprocal for negative exponents: Negative exponents indicate reciprocal of the base raised to the positive exponent.
  • Asymptote: The x-axis acts as a horizontal asymptote that the graph approaches but never crosses.

Examples

  1. Growth Example: ( y = 10 \cdot 3^x )
  2. Decay Example: ( y = 10 \cdot 0.5^x )
    • Observations: Graph approaches the x-axis, doesn’t cross it.

Analysis

  • Exponential Growth: ( b = 3 ) indicating growth.
  • Exponential Decay: ( b = 0.5 ) indicating decay.
  • Y-axis Asymptote: Referred to as "asymptote"; graph approaches but doesn’t cross.

Graphing Steps for Exponential Functions

  1. Identify the base ( b ) to determine growth or decay.
  2. Verify the initial value or y-intercept ( a ).
  3. Sketch the graph showing the y-intercept and whether the curve is increasing or decreasing.
  4. Label the horizontal asymptote.
  5. Determine domain and range:
    • Domain: All real numbers.
    • Range: Values ( y > 0 ).
  6. Use set builder notation for range.

Application in Word Problems

  • Growth Factor Identification: Convert percentage increases/decreases to decimal growth factors.
  • Example: Increase by 14% means multiply by 1.14.
  • Example Problem: Calculate when a 1959 Gibson L Paul guitar purchased for $12,000 will be valued at $60,000 if it increases by 14% per year.
    • Set up the equation: ( y = 12,000 \cdot 1.14^x )
    • Solve graphically: Find intersection with horizontal line ( y = 60,000 ).

Homework

  • Practice problems to graph and analyze exponential functions.
  • Explore the implications of growth and decay in real-world contexts.

These notes summarize the key points of exponential functions and their graphs, providing a foundational understanding of growth and decay processes in mathematics.