Math Example--Rational Concepts--Graphs of Rational Functions: Example 12

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x + 1) / ((x + 2)(x + 4)). The graph features two vertical asymptotes at x = -2 and x = -4, and displays three distinct branches, with one in each of the first, second, and third quadrants. Students are asked to graph the function and identify its vertical asymptotes.

Rational functions are a fundamental topic in algebra, involving fractions of polynomial functions. This collection of examples facilitates teaching the topic by providing visual representations of various rational functions, enabling students to observe how changes in the function affect its graph. By studying multiple examples, students can identify patterns and develop a deeper understanding of asymptotes and function behavior.

Exposure to multiple worked-out examples is crucial for students to fully grasp mathematical concepts. It allows them to encounter different scenarios, reinforces key ideas, and helps develop problem-solving skills. This repetitive exposure can boost confidence and enhance overall comprehension of rational functions.

Teacher's Script

As we examine this graph, notice the two vertical asymptotes at x = -2 and x = -4. These occur where the denominator equals zero. Observe how the function behaves differently near each asymptote and how it creates three distinct branches. Pay attention to how the function approaches positive or negative infinity as it nears each asymptote. Understanding these complex behaviors and their relationship to the function's equation is key to mastering rational functions with multiple factors in the denominator.

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