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Chapter 10: Problem 76

Find the asymptotes of the graph of each equation. $$ y=\frac{2}{x-3}-1 $$

Short Answer

Expert verified
The vertical asymptote of the function is \(x = 3\) and the horizontal asymptote is \(y = 0\).

Step by step solution

01

Identify the Vertical Asymptote

To find the vertical asymptote, we need to find the x-value that will make the function undefined. This is where the denominator equals zero. In this case, set the denominator \(x - 3\) equal to zero and solve: \(x - 3 = 0\). Solving this gives \(x = 3\), which is the vertical asymptote.
02

Identify the Horizontal Asymptote

The horizontal asymptote of a rational function can be found by taking the degree (the highest exponent) of the denominator and the numerator. If the degree of the denominator is larger than the numerator, then y=0 is our horizontal asymptote. Here, the degree of the numerator is 0 and the degree of the denominator is 1 since the numerator is 2 (a constant, with degree 0) and the denominator is \(x - 3\) (a linear term, and thus of degree 1). Hence, y=0 is our horizontal asymptote.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Vertical Asymptotes
Vertical asymptotes occur in rational functions where the function becomes undefined. This happens when the denominator equals zero, causing the function to "shoot off" to infinity or negative infinity. In simpler terms, vertical asymptotes represent values of x that a graph will never touch or cross.

For the function \( y = \frac{2}{x-3} - 1 \), the vertical asymptote is determined by setting the denominator \( x - 3 \) to zero: \( x - 3 = 0 \). Solving gives us \( x = 3 \).

The role of the vertical asymptote is crucial as it indicates a boundary for the graph, suggesting a sharp change on the visual representation.
Exploring Horizontal Asymptotes
Horizontal asymptotes help us understand how a function behaves as x approaches infinity or negative infinity. They show the y-value that the function approaches but may not necessarily touch.

In rational functions, horizontal asymptotes depend on the degree (highest power) of the numerator and the denominator. In our equation \( y = \frac{2}{x-3} - 1 \), the numerator is a constant (degree 0) and the denominator is linear (degree 1).

When the degree of the denominator is greater than the numerator, as here, the horizontal asymptote is \( y = 0 \). This tells us that the graph will level out at y=0 as x grows larger in the positive or negative direction.
Rational Functions Simplified
Rational functions are expressions that involve ratios of polynomials, akin to fractions but with variables. These functions are versatile and appear in many mathematical contexts. They are written as \( \frac{a(x)}{b(x)} \), where \( a(x) \) and \( b(x) \) are polynomials, and they can model everyday scenarios in engineering, economics, and beyond.

In our example \( y = \frac{2}{x-3} - 1 \), \( 2 \) is the numerator and \( x-3 \) is the denominator, creating a rational fraction. A key feature of these functions is their potential to have asymptotes, holes, or intercepts, which influences their graphs significantly.
Techniques for Graphing Equations
Graphing equations of rational functions can reveal much about their behavior. The process involves identifying key characteristics such as intercepts and asymptotes. These guide how the curve visually appears on a graph.

To graph \( y = \frac{2}{x-3} - 1 \):
  • First, find intercepts by setting each respective variable to zero.
  • Next, mark out the asymptotes, transitioning your attention from algebraic solutions to visual representation.
  • Then, evaluate other potential features, like symmetry or transformations.
This systematic approach helps create a detailed and accurate illustration of the function's behavior, making intricate relationships within the equation clearer.

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Most popular questions from this chapter

Find the foci for each equation of an ellipse. Then graph the ellipse. $$ \frac{x^{2}}{256}+\frac{y^{2}}{121}=1 $$

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