Problem 74 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=-\frac{1}{x-1} $$

Step-by-Step Solution

Answer

The asymptotes of the graph of the equation \(y= -1/(x-1)\) are \(x = 1\) (vertical asymptote) and \(y = 0\) (horizontal asymptote).

Step 1: Find the Vertical Asymptote

The vertical asymptote of the equation is the value of x that makes the denominator zero. To find it, set the denominator \(x-1\) equal to zero and solve for x. Thus, \(x-1 = 0\) leads to \(x = 1\), which is the vertical asymptote.

Step 2: Find the Horizontal Asymptote

The horizontal asymptote of an inverse variation equation like the one given is usually the y-value that the graph approaches as x goes to positive or negative infinity. For the equation \(y=-1/(x-1)\), note that as x becomes very large or very small, the value of y will approach zero. Hence, the horizontal asymptote is \(y = 0\).

Key Concepts

Vertical Asymptote

A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. For rational functions, vertical asymptotes indicate where the function is undefined. You can find these by setting the denominator equal to zero and solving for the variable. In the equation \(y=-\frac{1}{x-1}\), setting \(x-1 = 0\) solves to \(x = 1\). This tells us there is a vertical asymptote at \(x = 1\).

• The graph approaches \(x=1\) very closely but doesn't touch it.
• As you move toward this line on the graph from either side, the function's value becomes extremely large (positive or negative).

Understanding vertical asymptotes helps predict the behavior of graphs around undefined points.

Horizontal Asymptote

Horizontal asymptotes serve as a guideline for the end behavior of a graph. While a graph may touch or cross a horizontal asymptote, it ultimately returns to approach this line as \(x\) moves toward infinity or negative infinity. For the equation \(y=-\frac{1}{x-1}\), regardless of whether \(x\) becomes very large or very small, the value of \(y\) approaches zero. This is why the horizontal asymptote is \(y = 0\).

• Think of horizontal asymptotes as the 'leveling off' behavior of a function.
• They provide insight into how a function behaves at extreme values of \(x\) (both large positive and negative).

Recognizing horizontal asymptotes helps you understand long-term trends of rational functions.

Inverse Variation

Inverse variation describes a special relationship where the product of two variables is constant. In the function \(y=-\frac{1}{x-1}\), as \(x\) increases, \(y\) decreases proportionally, maintaining this relationship.

• Inverse variation contrasts with direct variation where two variables increase or decrease together.
• In equations, inverse variation often appears as \(y = \frac{k}{x}\), where \(k\) is a constant.

The concept helps in recognizing patterns where one quantity varies inversely as another, crucial in solving real-world problems involving speed, force, or pressure.

Rational Functions

A rational function is a ratio of two polynomials. These functions may have both vertical and horizontal asymptotes based on their equations. The function \(y=-\frac{1}{x-1}\) is a classic example of a rational function.

• They can show more complex behaviors like asymptotic behavior and special variation patterns.
• Understanding rational functions includes analyzing domains, asymptotes, intercepts, and limits.

Rational functions are essential in algebra, calculus, and applied sciences, where they model complex real-world processes and relationships.