Exponential Functions Definition and Characteristics

Exponential Functions Definition

• An exponential function is a function of the form f(x) = a^x, where 'a' is a positive constant (a > 0) and 'a' ≠ 1.
• The variable 'x' is the exponent.
• The base 'a' is the constant that is raised to the power of 'x'.
• The graph of an exponential function passes through the point (0, 1).
• Exponential functions are characterized by their rapid growth or decay.

Key Characteristics of Exponential Functions

• Growth: If 'a' is greater than 1 (a > 1), the function represents exponential growth. As 'x' increases, the function values increase significantly.
• Decay: If 'a' is between 0 and 1 (0 < a < 1), the function represents exponential decay. As 'x' increases, the function values decrease significantly.
• Horizontal Asymptote: All exponential functions (whether growth or decay) have a horizontal asymptote. This asymptote is typically the x-axis (y = 0), although it can shift vertically.
• Domain: The domain of an exponential function is always all real numbers.

Graphing Exponential Functions

• Graphing Steps: To graph an exponential function, plot points, and then connect them to form a smooth curve. This requires careful selection of x-values, especially if a or x are not simple values.
• Transformation of Functions: Like other functions, exponential functions can be transformed. Transformations such as vertical shifts, horizontal stretches or compressions, and reflections about the x-axis or y-axis can change the graph's appearance.

Exponential Growth Formula

• The general formula for exponential growth is: f(t) = a * b^t
  • Where:
    • f(t) represents the final amount.
    • a is the initial amount or principal.
    • b is the growth factor, and b > 1.
    • t is the time.

Exponential Decay Formula

• The general formula for exponential decay is: f(t) = a * b^t
  • Where:
    • f(t) represents the final amount.
    • a is the initial amount or principal.
    • b is the decay factor, and 0 < b < 1.
    • t is the time.

Compound Interest

• Compound interest is a common example of exponential growth.
• This is calculated using a formula that considers the initial principal, interest rate, number of times the interest is compounded per year, and the number of years the money is invested.
• The more frequently the interest compounds, the faster the investment grows.

Applications of Exponential Functions

• Finance: Compound interest calculations, loan amortization, and other financial models.
• Biology: Population growth models, radioactive decay, and bacterial growth.
• Physics: Radioactive decay, heat transfer, and other natural phenomena.
• Technology: Data modeling of technological changes, network growth and many other phenomena related to growth.

Important Concepts

• Growth Factor: The factor by which the initial value is multiplied in each successive time period.
• Decay Factor: The factor by which the initial value is multiplied in each successive time period.
• Continuous Growth: A special case of exponential growth that happens continuously rather than at discrete intervals. Continuous growth is often calculated using the natural exponential function, e^kt, where 'e' is Euler's number (approximately 2.718).