Exponential Functions Study Notes

Overview of Exponential Functions

General Structure of an Exponential Function

  • Function Form: The general equation is given by ( y = a(1 + b)^{2x} + k )
    • Where:
    • (a): Coefficient that affects the vertical stretch.
    • (b): Rate of growth (0.4 in this case).
    • (k): Vertical shift of the graph.

Key Features of the Equation ( y = a(1 + 0.4)^{2x} + k )

  1. Domain:

    • ( R ) (all real numbers)

  2. Range:

    • ( y > -1 ) (the function approaches but never reaches -1)

  3. y-intercept:

    • At ( x = 0 ): ( (0, 1) )
    • Calculation: [ y = a(1 + 0.4)^{2(0)} + k = 1 ]

  4. Horizontal Asymptote:

    • ( y = -1 ) (as ( x ) approaches ( - \infty ) it approaches this line)

  5. Behavior:

    • Increasing or Decreasing:
      • Function always increases as ( x ) approaches ( + \infty )
    • Left-End Behavior:
      • As ( x \to -\infty ), ( y \to -1 )
    • Right-End Behavior:
      • As ( x \to +\infty ), ( y \to +\infty )

Solving for Constants

  1. Find the value of k:

    • Utilize the equation ( y - k = 0.521 )
    • Calculation:
      • [ -0.479 - k = 0.521 ]
      • [ -0.479 - 0.521 = k ]
      • [ k = -1 ]

  2. Find the constant percent rate of change:

    • Given as 40% (which corresponds to the growth factor of 1.4 as the original rate is ( 1 + 0.4 )).

  3. Determine the value of a:

    • Rewrite the equation using the found k:
    • [ 1 = a(1.4)^{2(0)} - 1 ]
    • Simplifying gives:
      • [ 1 = a - 1 ]
      • [ a = 2 ]

Writing the Function in Growth Factor Form

  1. Convert ( y = a(1 + 0.4)^{2x} + k ) using found values:
    • [ y = 2(1 + 0.4)^{2x} - 1 ]
    • Which simplifies to:
    • [ y = 2(1.4)^{2x} - 1 ]

Exponential Function F(x)

  • Graph and Table of Values referenced but not detailed here.
    • Values such as ( (3, 7, 3, 898) ) indicate sample output.

Key Features of F(x)

  1. Domain:

    • Reference Domain.

  2. y-intercept:

    • Calculate using the values from the graph.

  3. Range:

    • Defined by the output values.

  4. Horizontal Asymptote:

    • Identical to previous features, indicating behavior at extremes.

  5. Increasing or Decreasing:

    • Clarify behavior based on values at known points.

  6. Left-End and Right-End Behavior:

    • Based on asymptotes and the sign of the growth rate.

Constants for G(x)

  • General Structure: The equation is described as ( y = a(10^{0.15}) + k )

  1. Specify Domain, Range, y-intercept, and Asymptote:

    • Calculate using observed data in graphs and tables.

  2. Determine the value of k and a:

    • Utilize equations and point values to solve for these constants.

  3. Calculate Growth Factor:

    • Essential characteristic of exponential functions; derive from established elements.

  4. Write G(x) in Growth Factor Form:

    • Express function in a direct way that highlights growth behavior.