Provide a short answer to each question. What is the equation of the vertical asymptote of the graph of y=[1/(x-3)]+2? Of the horizontal asymptote?

Verified step by step guidance

1. Identify the vertical asymptote by finding the value of \(x\) that makes the denominator zero. For the function \(y=\frac{1}{x-3}+2\), set the denominator equal to zero: \(x - 3 = 0\).
2. Solve the equation \(x - 3 = 0\) to find the vertical asymptote: \(x = 3\).
3. To find the horizontal asymptote, analyze the behavior of the function as \(x\) approaches infinity or negative infinity. The term \(\frac{1}{x-3}\) approaches zero as \(x\) becomes very large or very small.
4. Since \(\frac{1}{x-3}\) approaches zero, the function \(y = \frac{1}{x-3} + 2\) approaches \(y = 2\) as \(x\) goes to infinity or negative infinity.
5. Therefore, the horizontal asymptote is the line \(y = 2\).

Key Concepts

Vertical Asymptote

A vertical asymptote occurs where the function approaches infinity or negative infinity as the input approaches a specific value, typically where the denominator of a rational function is zero. For y = 1/(x-3) + 2, the vertical asymptote is found by setting the denominator (x-3) equal to zero, giving x = 3.

Horizontal Asymptote

A horizontal asymptote describes the behavior of a function as the input approaches positive or negative infinity. It represents a constant value that the function approaches but does not necessarily reach. For rational functions like y = 1/(x-3) + 2, the horizontal asymptote is the constant term added outside the fraction, here y = 2.

Rational Functions and Their Graphs

Rational functions are ratios of polynomials and often have asymptotes where the denominator is zero or at infinity. Understanding how to analyze their graphs involves identifying points of discontinuity (vertical asymptotes) and end behavior (horizontal asymptotes), which helps in sketching and interpreting the function's behavior.