Math Example--Rational Concepts--Rational Functions in Tabular and Graph Form: Example 20

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (x - 1) - 1. The image showcases both the table of x-y coordinates and the resulting graph. The table contains specific x-y coordinates: (-2, -1.333), (-1.5, -1.4), (0, -2), (1.5, 1), and (2, 0). The graph is a hyperbola with shifted asymptotes at x = 1 and y = -1, illustrating how subtracting a constant in the denominator and from the entire fraction affects the graph's shape and position.

Rational functions are an important concept in algebra and calculus, representing the ratio of two polynomial functions with additional transformations. This collection of examples aids in teaching this topic by providing visual representations of various rational functions, allowing students to observe how alterations in the function impact its graph. By studying multiple examples, students can identify patterns and develop a comprehensive understanding of how constants in different parts of the function shift the graph both horizontally and vertically.

The importance of presenting multiple worked-out examples cannot be overstated when it comes to students fully grasping the concept of rational functions and their transformations. Each example builds upon the previous ones, introducing slight variations in the function that result in different graph shapes and positions. This approach enhances students' pattern recognition skills and improves their ability to predict the behavior of rational functions based on their equations.

Teacher's Script: Now, let's analyze our twentieth example: y = 1 / (x - 1) - 1. How does this function compare to our previous example? Notice that we've kept the subtraction in the denominator but now we're subtracting 1 from the entire fraction instead of adding it. What effect do you think this change will have on the graph? As you observe the graph, can you explain why the asymptotes are at x = 1 and y = -1? How does it compare to y = 1 / (x - 1) + 1? As we continue exploring more examples, try to predict how different combinations of constants will affect the graphs.

For a complete collection of math examples related to Rational Functions click on this link: Math Examples: Rational Functions in Tables and Graphs Collection.