Understanding Exponential Functions

Key Components:

• Base (b): The constant multiplier in the form b^x.
• Exponent (x): The variable that represents the power to which the base is raised.
• Initial Value (a): The value of the function when x = 0.

Graphing Exponential Functions:

• To graph an exponential function, choose a set of x-values, calculate the corresponding y-values using the function, and plot the points on a graph.
• Exponential functions can have different shapes based on the value of the base "b" and whether the function represents growth or decay.

Properties of Exponential Functions:

• Exponential functions with a base greater than 1 represent growth, while those with a base between 0 and 1 represent decay.
• Exponential functions never cross the x-axis, as they are always positive for real values of x.
• Exponential growth functions increase at an increasing rate, while exponential decay functions decrease at a decreasing rate.

Applications of Exponential Functions:

• Modeling population growth or decay.
• Describing the decay of radioactive substances.
• Calculating compound interest in finance.

Practice Problems:

1. Graph the function f(x) = 2^x.
2. Determine whether the function g(x) = (1/3)^x represents growth or decay.
3. If an investment grows at a rate of 5% per year, write an exponential function to model the growth of the investment over time.