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. Step 1: Identify the form of the function. The given function is y = \frac{1}{x-3} + 2, which is a rational function.
2. Step 2: Determine the vertical asymptote. The vertical asymptote occurs where the denominator is zero. Set x - 3 = 0 and solve for x.
3. Step 3: Solve for the vertical asymptote. Add 3 to both sides of the equation to find the value of x.
4. Step 4: Determine the horizontal asymptote. For the function y = \frac{1}{x-3} + 2, as x approaches infinity, the term \frac{1}{x-3} approaches 0.
5. Step 5: Identify the horizontal asymptote. The horizontal asymptote is the constant term added to the function, which is y = 2.

Key Concepts

Vertical Asymptote

A vertical asymptote occurs in a rational function when the denominator approaches zero, causing the function to approach infinity. For the function y = 1/(x-3) + 2, the vertical asymptote is found by setting the denominator equal to zero, which gives x = 3. This means that as x approaches 3, the value of y will increase or decrease without bound.

Horizontal Asymptote

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. In the case of y = 1/(x-3) + 2, as x approaches infinity, the term 1/(x-3) approaches 0, leading to a horizontal asymptote at y = 2.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors such as asymptotes, which are lines that the graph approaches but never touches. Understanding the structure of rational functions is essential for analyzing their graphs, including identifying vertical and horizontal asymptotes, which are critical for sketching the function accurately.