Math Example--Rational Concepts--Graphs of Rational Functions: Example 18

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x - 8) / ((x + 3)(x - 3)). The graph features vertical asymptotes at x = -3 and x = 3, highlighted by dashed red lines. Students are asked to graph the function and identify its vertical asymptotes.

Rational functions are a significant 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

Let's analyze this graph carefully. Notice the two vertical asymptotes at x = -3 and x = 3, highlighted by 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, especially the constant term -8, influences the overall shape and position of the graph. Understanding these complex behaviors and their relationship to the function's equation is essential 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.