Exponential and Logarithmic Functions

Exponential Functions

  • Definition of Exponential Function:

    • An exponential function with base b is defined as:

    • f(x) = b^x or y = b^x

    • Key points:

    • b must be a positive constant (b > 0) and b cannot be 1.

    • x can be any real number.

  • Examples of Exponential Functions:

    • Functions of the form:

    • y = 2^x, y = 3^x, y = {e}^x

    • Characteristics:

    • Base must be greater than 1.

Non-examples of Exponential Functions

  • Functions that do not qualify as exponential:

    • Variable in base (e.g., x^2)

    • Base = 1 (e.g., 1^x = 1)

    • Negative base (e.g., -2^x)

    • Variable in both base and exponent (e.g., x^x)

Using a Calculator

  • Calculating Exponential Values:

    • Example: Calculate e^(3/4) using the calculator:

    • Use the second key and ln button to access e^x.

    • Input the exponent (3/4) and hit enter.

    • Round the result to three decimal places.

  • Sample calculations:

    • e^(3/4) = 2.117

    • 3^(2.4) = 13.967

    • 5^(√3) = 16.242 (not shown in detail but follow similar steps as above)

Graphing Exponential Functions

  • Basic Graphing Procedure:

    • Start with values of x (e.g., -2, -1, 0, 1, 2) and compute corresponding y values using y = b^x.

    • Identify key points and plot them on a graph.

  • Example of Graphing y = 2^x:

    • x = -2, y = 1/4

    • x = -1, y = 1/2

    • x = 0, y = 1

    • x = 1, y = 2

    • x = 2, y = 4

  • Properties of Graphs:

    • Horizontal Asymptote: y = 0 (the graph approaches but never touches the x-axis).

    • Domain: All real numbers (-∞, ∞).

    • Range: (0, ∞). Can never be negative.

Characteristics of Exponential Functions

  • Key Features:

    • Domain: All real numbers (-∞, ∞).

    • Range: (0, ∞) (y cannot be zero or negative).

    • Intercepts: (0, 1) - y-intercept only, no x-intercept.

    • Behavior:

    • If b > 1, function increases (upwards to the right).

    • If 0 < b < 1, function decreases (downwards to the right).

    • It's a One-to-One Function (passes both vertical and horizontal line tests).

Transformations of Exponential Graphs

  • Transformations:

    • Vertical translation:

    • If f(x) is in the form f(b^x + C), it moves up C units (if C > 0) or down C units (if C < 0).

    • Horizontal translation:

    • If it’s f(b^(x + C)), it moves left C units (if C > 0) or right C units (if C < 0).

    • Reflections:

    • If the negative sign is not affecting x, it reflects across the x-axis.

    • If the negative sign is with x, it flips across the y-axis.

    • Stretching/Shrinking:

    • If not with x, multiply y by a factor C. Greater than 1 = stretch, between 0 and 1 = shrink.

      • If with x, use 1/C to stretch/shrink.

Examples of Graphing with Transformations

  1. Graph y = 3^(x + 1):

    • Shift left 1 unit.

  2. Graph h(x) = 2^x - 3:

    • Shift down 3 units and the new horizontal asymptote becomes at y = -3.

  3. Combined transformations: e.g., h(x) = 2^(x - 2) + 1:

    • Shift right 2 units and up 1 unit, adjust the horizontal asymptote accordingly.

Conclusion

  • Important Points to Remember:

    • Know the definitions and properties of exponential functions.

    • Be familiar with using graphing calculators for calculations and graphing.

    • Understand and correctly apply transformation rules for accurate graphing of exponential functions.