Exponential Functions and Decay Concepts

The video tutorial explains exponential decay using the example of cutting a string in half repeatedly. It introduces exponential functions, focusing on the form y = b^x, and discusses the characteristics of exponential decay when the base is between 0 and 1. The tutorial models this concept by cutting a string and organizing the data in a table, showing how the length decreases exponentially. It also covers graphing these functions, highlighting the horizontal asymptote at y = 0 and the y-intercept at (0,1). The video concludes by summarizing the key features of exponential decay functions.

What happens to the length of a string after 16 cuts if you start with a 1-mile long string?

• It becomes 1 inch long.

• It remains 1 mile long.

• It becomes 1 foot long.

• It becomes 1 yard long.

Which of the following is a characteristic of an exponential decay function?

• The base is greater than 1.

• The base is between 0 and 1.

• The base is equal to 1.

• The base is a negative number.

How can you express the function y = 1/3^(-x) in terms of exponential growth?

• y = 1/3^x

• y = 3^x

• y = 3^(-x)

• y = 1/2^x

What pattern do the denominators follow when cutting a string in half repeatedly?

• Powers of 3

• Powers of 2

• Powers of 5

• Powers of 10

What is the y-intercept of the exponential function y = (1/2)^x?

• (0, 0)

• (1, 1)

• (1, 0)

• (0, 1)

What happens to the value of y as x approaches infinity in an exponential decay function?

• y approaches 1

• y approaches infinity

• y becomes negative

• y approaches 0

What is a horizontal asymptote in the context of exponential functions?

• A diagonal line

• A vertical line

• A line that the graph crosses

• A line that the graph never touches

What is the general shape of the graph of an exponential function?

• A straight line

• A smooth curve

• A circle

• A zigzag line

What is the behavior of the graph of an exponential decay function as it approaches the x-axis?

• It becomes parallel to the x-axis

• It moves away from the x-axis

• It crosses the x-axis

• It approaches but never touches the x-axis

What is a key characteristic of all exponential decay functions?

• They have a vertical asymptote at x = 0

• They have a horizontal asymptote at y = 0

• They increase rapidly

• They have a y-intercept at (1, 0)