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Asymptote

Asymptotes are lines that a graph approaches but never touches. In other words, as the x or y value of a graph becomes larger and larger, the distance between the graph and the asymptote becomes smaller and smaller, but they never actually meet. Asymptotes can be horizontal, vertical, or slanted.

There are three types of asymptotes: horizontal, vertical, and oblique.

A horizontal asymptote is a horizontal line that the graph approaches as the x value becomes very large. For example, the graph of the function y = 1/x has a horizontal asymptote at y = 0 because as x becomes very large, the value of 1/x approaches 0.

A vertical asymptote is a vertical line that the graph approaches but never touches as the y value becomes very large. For example, the graph of the function y = x^2 – 4x + 5 has a vertical asymptote at x = 2 because as y becomes very large, the value of x approaches 2.

An oblique asymptote is a slanted line that the graph approaches but never touches as the x value becomes very large. For example, the graph of the function y = x^2 + 4x + 5 has an oblique asymptote at y = x because as x becomes very large, the value of x^2 becomes much larger than the value of 4x and 5, so the graph approaches the line y = x.

Examples:

1. The graph of the function y = 1/x has a horizontal asymptote at y = 0 because as x becomes very large, the value of 1/x approaches 0.
2. The graph of the function y = x^2 – 4x + 5 has a vertical asymptote at x = 2 because as y becomes very large, the value of x approaches 2.
3. The graph of the function y = x^2 + 4x + 5 has an oblique asymptote at y = x because as x becomes very large, the value of x^2 becomes much larger than the value of 4x and 5, so the graph approaches the line y = x.
4. The graph of the function y = (x-1)/(x^2+1) has a vertical asymptote at x = 1 because as y becomes very large, the value of x approaches 1.
5. The graph of the function y = x^3 – 3x^2 + x has an oblique asymptote at y = x because as x becomes very large, the value of x^3 becomes much larger than the value of 3x^2 and x, so the graph approaches the line y = x.

Quiz:

1. Which of the following is NOT a type of asymptote? a) horizontal b) vertical c) slanted d) circular

Answer: d) circular

2. Which of the following statements about asymptotes is NOT true? a) Asymptotes are lines that a graph approaches but never touches. b) As the x or y value of a graph becomes larger and larger, the distance between the graph and the asymptote becomes smaller and smaller. c) Asymptotes can be horizontal, vertical, or slanted. d) Asymptotes are always at a fixed distance from the graph.

Answer: d) Asymptotes are always at a fixed distance from the graph.