Math Example--Rational Concepts--Graphs of Rational Functions: Example 17

Topic

Rational Functions

Description

This example presents the graph of the rational function y = (x - 1) / ((x - 2)(x - 3)). The graph features vertical asymptotes at x = 2 and x = 3, marked with dashed red lines. Students are tasked with graphing the function and determining its vertical asymptotes.

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

Exposure to multiple worked-out examples is essential for students to fully comprehend mathematical concepts. It enables them to encounter different scenarios, reinforces key ideas, and helps develop problem-solving skills. This repetitive exposure can increase confidence and enhance overall understanding of rational functions.

Teacher's Script

Let's analyze this graph carefully. Notice the two vertical asymptotes at x = 2 and x = 3, clearly marked with dashed red lines. These occur where the denominator equals zero. Observe how the function approaches positive or negative infinity as it nears each asymptote. Pay attention to how the numerator, being similar to the factors in the denominator, affects the overall shape of the graph. Understanding these complex behaviors and their relationship to the function's equation is crucial for 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.