Horizontal and Vertical Asymptotes - Slant / Oblique - Holes - Rational Function - Domain & Range

TL;DR

This video explains how to identify and graph the vertical, horizontal, and slant asymptotes of rational functions.

Transcript

in this video we're going to focus on finding the horizontal and vertical asymptote of a rational function in addition we're also going to find the slant or oblique asymptote if it's there so let's begin let's go over our first function y is equal to one over x minus three to find the vertical asymptote set the denominator equal to zero so if we se... Read More

Key Insights

• 😫 Finding the vertical asymptote involves setting the denominator equal to zero.
• 🚥 Determining the horizontal asymptote depends on the degrees of the numerator and denominator.
• 😥 Graphing rational functions involves considering the behavior near asymptotes and test points on either side.
• 🧡 The domain is determined by removing the vertical asymptote, and the range by removing the horizontal asymptote.

Questions & Answers

Q: How do you find the vertical asymptote of a rational function?

To find the vertical asymptote, set the denominator equal to zero and solve for x. The result is the x-value of the vertical asymptote.

Q: What does it mean for a function to be "bottom heavy" when determining the horizontal asymptote?

A function is considered bottom heavy when the degree of the denominator is higher than that of the numerator. This means that the horizontal asymptote is y = 0.

Q: How do you determine the end behavior of a function?

The end behavior of a function is determined by examining the behavior of the function as x approaches positive and negative infinity. In this case, as x approaches positive infinity, the function approaches the horizontal asymptote, and as x approaches negative infinity, it also approaches the horizontal asymptote.

Q: How is the domain and range of a rational function determined?

The domain represents all the possible values that x can take, excluding the vertical asymptote. The range represents all the possible values that y can take, excluding the horizontal asymptote.

Summary & Key Takeaways

• Vertical asymptotes can be found by setting the denominator equal to zero. The vertical asymptote for y = 1/(x-3) is x = 3.

• Horizontal asymptotes are determined by the degrees of the numerator and denominator. The horizontal asymptote for y = 1/(x-3) is y = 0.

• To graph the function, test points on either side of the asymptotes are used. For y = 1/(x-3), the graph is above the horizontal asymptote on the right side and below it on the left side.