📘intermediate algebra review

key term - Exponential Function

Definition

An exponential function is a mathematical function in which the independent variable appears as the exponent. These functions exhibit a characteristic growth or decay pattern, where the rate of change is proportional to the current value of the function. Exponential functions are widely used to model a variety of real-world phenomena, from population growth to radioactive decay.

5 Must Know Facts For Your Next Test

  1. Exponential functions are characterized by the general form $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the base of the exponential function.
  2. When $b > 1$, the function exhibits exponential growth, and when $0 < b < 1$, the function exhibits exponential decay.
  3. The graph of an exponential function is a curved line that either increases or decreases at an accelerating rate, depending on the value of $b$.
  4. Logarithmic functions are the inverse of exponential functions, allowing for the transformation of exponential equations into linear equations.
  5. Exponential functions are essential in modeling various phenomena, such as population growth, radioactive decay, compound interest, and the spread of infectious diseases.

Review Questions

  • Explain how the graph of an exponential function differs from the graph of a linear function.
    • The graph of an exponential function is a curved line that either increases or decreases at an accelerating rate, whereas the graph of a linear function is a straight line with a constant rate of change. Exponential functions exhibit a characteristic growth or decay pattern, where the rate of change is proportional to the current value of the function. This results in a graph that is concave upward for exponential growth and concave downward for exponential decay, in contrast to the constant slope of a linear function.
  • Describe the role of exponential functions in modeling real-world phenomena, such as population growth or radioactive decay.
    • Exponential functions are widely used to model a variety of real-world phenomena that exhibit a characteristic growth or decay pattern. For example, in the case of population growth, the rate of change in population size is proportional to the current population, leading to an exponential growth pattern. Similarly, in radioactive decay, the rate of decay of a radioactive substance is proportional to the amount of the substance present, resulting in an exponential decay function. These models allow for the accurate prediction and analysis of these processes, which are essential in fields such as biology, physics, and economics.
  • Explain how the properties of logarithms can be used to solve exponential equations and transform exponential functions into linear functions.
    • Logarithmic functions are the inverse of exponential functions, allowing for the transformation of exponential equations into linear equations. By applying the properties of logarithms, such as the power rule ($\log_b(x^y) = y \log_b(x)$) and the logarithm of a product ($\log_b(xy) = \log_b(x) + \log_b(y)$), exponential equations can be rewritten in a linear form. This transformation enables the use of linear algebra techniques to solve exponential equations and simplify the analysis of exponential functions. The ability to convert exponential functions into linear functions is particularly useful in various applications, such as in the analysis of compound interest, population growth, and radioactive decay.