Graphing Rational Functions With Vertical, Horizontal & Slant Asymptotes, Holes, Domain & Range

TL;DR

Learn how to graph rational functions by identifying the asymptotes, holes, and determining the domain and range.

Transcript

in this video we're going to focus on graphing rational functions we're going to talk about how to graph the asymptotes horizontal vertical even the slant and oblique asymptotes how to define the holes and also how to write the domain and range of the function so let's go ahead and begin let's start with the parent function Y is equal to 1 over X n... Read More

Key Insights

• 🚥 Graphing rational functions involves identifying and graphing both vertical and horizontal asymptotes.
• ☺️ The domain of a rational function excludes the vertical asymptotes and x-coordinate of any holes.
• 😀 The range of a rational function excludes the horizontal asymptote and y-coordinate of any holes.
• ❣️ Graphs of rational functions can exhibit symmetry about the y-axis or x-axis, depending on the parent function.
• ❣️ Rational functions with a bottom-heavy denominator have a horizontal asymptote at y = 0.

Questions & Answers

Q: What is the domain and range of the function Y = 1/x?

The domain is from negative infinity to 0, union 0 to positive infinity, excluding x = 0. The range is from negative infinity to 0, union 0 to positive infinity, excluding y = 0.

Q: How do you graph a rational function with a vertical asymptote at x = 1 and a horizontal asymptote at y = 0?

Start by plotting the vertical asymptote at x = 1 and the horizontal asymptote at y = 0. Then, plot points to the left and right of the asymptote to sketch the graph accurately.

Q: How does the graph of Y = -1/x differ from the graph of Y = 1/x?

The graph of Y = -1/x is the reflection of the graph of Y = 1/x over the x-axis. It is symmetric about the x-axis and exists in the upper left and lower right corners.

Q: How can you graph a rational function with a slant asymptote?

To find a slant asymptote, perform long division on the function. The resulting quotient will be the equation of the slant asymptote. Plot the asymptote and points around it to sketch the graph accurately.

Summary & Key Takeaways

• Graphing rational functions involves identifying the vertical and horizontal asymptotes, determining the domain and range, and finding any holes in the graph.

• The parent function of Y = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The graph exists in the upper right and lower left corners.

• For Y = 1/x^2, the parent function has the same vertical asymptote at x = 0 but the horizontal asymptote is y = 0. The graph is above the x-axis and symmetric about the y-axis.