Exponential Functions Definition and Characteristics

Study Notes

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).

Description

This quiz covers the definition and key characteristics of exponential functions, including their forms and properties of growth and decay. Gain a deeper understanding of how these functions behave and their graphical representation as you explore their unique features.