The Graphs of y = 1/x and y = 1/x^2 College Algebra

TL;DR

This video explains the graphs of two important mathematical functions: y = 1/x and y = 1/x^2.

Transcript

in this video we're going to talk about two special functions so y equals one over x and y equals one over x squared these are both important functions in mathematics and they come up a lot so it's important to be familiar with their graphs so first we'll draw 1 over X there's the y-axis there's the x-axis so X Y and this function has a horizontal ... Read More

Key Insights

• ❣️ The functions y = 1/x and y = 1/x^2 are important in mathematics and have distinct graph characteristics.
• ❣️ The graph of y = 1/x approaches a horizontal asymptote at y = 0 and has a vertical asymptote.
• ❣️ The graph of y = 1/x^2 resembles a volcano bowl and has a horizontal asymptote on the x-axis and a vertical asymptote on the y-axis.
• ↔️ Translating these graphs involves shifting them left, right, up, or down based on the given equation.
• 🪜 Adding or subtracting values to/from the equation affects the direction and magnitude of the translation.
• 🚥 The vertical asymptote remains unchanged during translation, while the horizontal asymptote shifts based on the translation direction.
• ❣️ The functions y = 1/x and y = 1/x^2 are frequently encountered in various mathematical applications.

Questions & Answers

Q: What are the key characteristics of the function y = 1/x?

The function y = 1/x has a horizontal asymptote at y = 0 and a vertical asymptote. Its graph never crosses the vertical asymptote and approaches but never touches the asymptotes.

Q: How is the function y = 1/x^2 different from y = 1/x?

The function y = 1/x^2 is a version of y = 1/x squared. It has a horizontal asymptote on the x-axis and a vertical asymptote on the y-axis. The graph looks like a volcano bowl.

Q: How does adding or subtracting values to/from the equation affect the graph?

Adding or subtracting values from the equation shifts the graph in different directions. Adding values to the x-axis shifts the graph to the left, and subtracting values from the x-axis shifts it to the right. Adding values to the entire function shifts it upward, and subtracting values shifts it downward.

Q: What happens to the vertical and horizontal asymptotes when translating the graph?

When translating the graph, the vertical asymptote remains unchanged, but the horizontal asymptote shifts based on the direction of the translation. Upward translations shift the horizontal asymptote upward, and downward translations do not affect the horizontal asymptote.

Summary & Key Takeaways

• The function y = 1/x has a horizontal asymptote at y = 0 and a vertical asymptote. The graph approaches but never touches these asymptotes.

• The function y = 1/x^2 is similar to y = 1/x, but it has a horizontal asymptote on the x-axis and a vertical asymptote on the y-axis. The graph looks like a volcano bowl.

• Translating these graphs involves shifting them left, right, up, or down based on the given equation.