Question

The graph of the rational function r is shown. Which of the following tables could be used to describe the asymptotic behavior of r at x = -1 and at

2=1?

Answer 1

To describe the asymptotic behavior of a rational function at x = -1 and x = 2, a table can be used to show x-values approaching -1 and 2, with the corresponding y-values that the function approaches. The table should contain values of x that are close to -1 and 2, and the corresponding y-values as x gets closer and closer to those points.

Step-by-step explanation:

The table that could be used to describe the asymptotic behavior of the rational function at x = -1 and x = 2 is a table that contains the values of x approaching -1 and 2 from both sides. The function will have horizontal asymptotes at y = some value when x approaches -1 and y = some value when x approaches 2. So, the table would show x-values that are close to -1 and 2, and the corresponding y-values that the function approaches as x gets closer and closer to -1 and 2.

For example, let's say the function approaches y = 3 when x gets close to -1 and y = -2 when x gets close to 2. The table could look like this:

x y

-0.9 2.8

-0.99 2.9

-0.999 2.99

-0.9999 2.999

1.9 -1.8

1.99 -1.9

1.999 -1.99

1.9999 -1.999

Answer 2

- For x = -1, if the values of y approach the same constant from both sides in any of the given tables, then the graph has a horizontal asymptote at that constant value.

- For x = 2, if the values of y approach the same constant from both sides in any of the given tables, then the graph has a horizontal asymptote at that constant value.

By analyzing the given tables and checking the values of y as x approaches -1 and 2 from both sides, you can determine which table(s) could be used to describe the asymptotic behavior of the rational function at x = -1 and x = 2.

Step-by-step explanation:

The graph of a rational function can provide information about its asymptotic behavior. To determine the asymptotic behavior of the function at x = -1 and x = 2, we need to analyze the end behavior of the graph.

When approaching x = -1 from the left side, if the function approaches a horizontal line, it means that the graph has a horizontal asymptote at y = a, where a is a constant. Similarly, when approaching x = -1 from the right side, if the function approaches a horizontal line, it means that the graph has a horizontal asymptote at y = b, where b is a constant.

To determine the behavior at x = -1, we can check the values of y as x approaches -1 from both sides using the given tables. If the values of y approach the same constant from both sides, then the graph has a horizontal asymptote at that constant value. If the values of y approach different constants from both sides, then the graph does not have a horizontal asymptote at x = -1.

Similarly, we can repeat this process to determine the behavior at x = 2. We check the values of y as x approaches 2 from both sides using the given tables. If the values of y approach the same constant from both sides, then the graph has a horizontal asymptote at that constant value. If the values of y approach different constants from both sides, then the graph does not have a horizontal asymptote at x = 2.