Math Example--Rational Concepts--Rational Functions in Tabular and Graph Form: Example 18

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x + 1) - 1. The image presents both the table of x-y coordinates and the resulting graph. Points plotted include (-2, -0.667), (-1.5, -0.6), (0, 0), (1.5, -3), and (2, -2). The graph shows a hyperbola with vertical and horizontal shifts, demonstrating how negating the x-term in the denominator and subtracting a constant affects the graph's shape and position.

Rational functions are a fundamental concept in algebra and calculus, representing the ratio of two polynomial functions with additional transformations. This collection of examples supports the teaching of this topic by providing visual representations of various rational functions, enabling students to observe how different combinations of negations and constants can result in diverse graph shapes and positions. By examining multiple examples, students can identify patterns and develop a deeper understanding of rational functions' behavior and transformations.

Presenting multiple worked-out examples is crucial for students to fully comprehend the concept of rational functions and their transformations. Each example builds upon the previous ones, introducing different combinations of negations and constants that result in unique graph shapes and positions. This approach enhances students' analytical skills and improves their ability to recognize and predict the effects of various transformations on rational functions.

Teacher's Script: Now, let's analyze our eighteenth example: y = 1 / (-x + 1) - 1. How does this function compare to our previous examples? Notice that we've kept the negated x-term in the denominator but subtracted 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 it's shifted both vertically and horizontally? How does it compare to y = 1 / (-x + 1) + 1? As we continue exploring more examples, try to predict how different combinations of negations and 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.