The Main Idea

• Exponential Regression:
  • Used when data grows or decays at an increasing rate
  • Model form: [latex]y = ab^x[/latex]
  • If [latex]b > 1[/latex]: exponential growth
  • If [latex]0 < b < 1[/latex]: exponential decay

Key Techniques

1. Using a Graphing Calculator:
   • Enter data in lists L1 (x-values) and L2 (y-values)
   • Create a scatter plot to verify exponential pattern
   • Use “ExpReg” from STAT CALC menu
   • Record values for [latex]a[/latex] and [latex]b[/latex]
2. Using Technology (Desmos):
   • Create a table with your data
   • Enter [latex]y_1 \sim ab^{x_1}[/latex] below the table
   • The tool displays values for [latex]a[/latex] and [latex]b[/latex]

The Main Idea

• Logarithmic Regression:
  • Used when data increases or decreases rapidly at first, then slows
  • Model form: [latex]y = a + b\ln(x)[/latex]
  • All input values must be positive
  • If [latex]b > 0[/latex]: increasing model
  • If [latex]b < 0[/latex]: decreasing model

Key Techniques

1. Using a Graphing Calculator:
   • Enter data in lists L1 and L2
   • Verify scatter plot shows logarithmic pattern
   • Use “LnReg” from STAT CALC menu
   • Record model as [latex]y = a + b\ln(x)[/latex]

The Main Idea

• Logistic Regression:
  • Used when growth starts rapidly then levels off to an upper limit
  • Model form: [latex]y = \frac{c}{1 + ae^{-bx}}[/latex]
  • [latex]c[/latex] is the limiting value (carrying capacity)
  • Initial value is [latex]\frac{c}{1+a}[/latex]

Key Techniques

1. Using a Graphing Calculator:
   • Enter data in lists L1 and L2
   • Verify scatter plot shows logistic pattern (S-shaped curve)
   • Use “Logistic” from STAT CALC menu
   • Record values for [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]