Y = b^x

Definition

The equation y = b^x represents an exponential function, where y is the dependent variable, b is the base of the exponential function, and x is the independent variable. This function describes a relationship where the output y grows or decays exponentially as the input x changes.

5 Must Know Facts For Your Next Test

1. The graph of the exponential function y = b^x is a curve that either increases or decreases depending on the value of the base b.
2. When the base b is greater than 1, the function exhibits exponential growth, meaning the output y increases rapidly as the input x increases.
3. When the base b is between 0 and 1, the function exhibits exponential decay, meaning the output y decreases rapidly as the input x increases.
4. The rate of growth or decay is determined by the value of the base b. The closer the base is to 1, the slower the rate of change.
5. Exponential functions are commonly used to model real-world phenomena, such as population growth, radioactive decay, and compound interest.

Review Questions

• Explain the relationship between the base b and the behavior of the exponential function y = b^x.
  • The base b in the exponential function y = b^x determines whether the function exhibits exponential growth or exponential decay. When b is greater than 1, the function will show exponential growth, meaning the output y increases rapidly as the input x increases. Conversely, when b is between 0 and 1, the function will show exponential decay, meaning the output y decreases rapidly as the input x increases. The closer the base is to 1, the slower the rate of change, either growth or decay.
• Describe how the graph of the exponential function y = b^x changes based on the value of the base b.
  • The graph of the exponential function y = b^x is a curve that either increases or decreases depending on the value of the base b. When b is greater than 1, the graph will be an increasing curve, exhibiting exponential growth. When b is between 0 and 1, the graph will be a decreasing curve, exhibiting exponential decay. The steepness of the curve is determined by the value of b, with larger values of b resulting in a steeper curve and faster growth or decay.
• Explain how exponential functions can be used to model real-world phenomena and the significance of this in various fields.
  • Exponential functions represented by the equation y = b^x are commonly used to model a wide range of real-world phenomena that exhibit growth or decay patterns. For example, exponential functions can be used to model population growth, radioactive decay, compound interest, and the spread of infectious diseases. The ability to accurately model these processes using exponential functions is crucial in fields such as biology, physics, finance, and epidemiology, as it allows for better understanding, prediction, and management of these important phenomena.

Related terms

Exponential Growth: Exponential growth is a type of function where the variable increases by a constant percent change over equal intervals of the independent variable.

Exponential Decay: Exponential decay is a type of function where the variable decreases by a constant percent change over equal intervals of the independent variable.

Base of the Exponential Function: The base, b, in the equation y = b^x, determines the rate of growth or decay of the exponential function. The base must be a positive real number, and values greater than 1 result in exponential growth, while values between 0 and 1 result in exponential decay.