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Chapter 1: Problem 47

Graph the functions. $$y=\frac{1}{x-2}.$$

Short Answer

Expert verified
The function has a vertical asymptote at \( x=2 \) and a horizontal asymptote at \( y=0 \). The graph is hyperbolic, with branches in the first and third quadrants.

Step by step solution

01

Understanding the function

The given function is \( y = \frac{1}{x-2} \). This is a rational function with a vertical asymptote at \( x = 2 \) and a horizontal asymptote at \( y = 0 \). The function is undefined at \( x = 2 \) because this value makes the denominator zero.
02

Identifying asymptotes

Identify the vertical asymptote by setting the denominator to zero: \( x - 2 = 0 \Rightarrow x = 2 \). The horizontal asymptote is \( y = 0 \) as \( x \to \pm \infty \), the term \( 1/(x-2) \to 0 \). These asymptotes guide the function's behavior and graph.
03

Plotting key points

Choose values for \( x \) around the vertical asymptote and substitute into the function to find corresponding \( y \) values. For example, if \( x = 1 \), then \( y = -1 \); if \( x = 3 \), then \( y = 1 \). These values show how the curve behaves around the asymptotes.
04

Sketching the graph

Start by sketching the asymptotes. Plot the points you calculated previously: \( (1, -1) \), \( (3, 1) \). Draw the curve that approaches the asymptotes. The graph will be in two parts, one in the first quadrant and another in the third quadrant, approaching the asymptotes but never touching them.

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Key Concepts

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

Vertical Asymptotes
Vertical asymptotes are lines that a graph approaches but never crosses or touches. They occur where a rational function's denominator is equal to zero. In the given function, \( y = \frac{1}{x-2} \), the denominator is \( x - 2 \). To find the vertical asymptote, we set the denominator to zero:
  • \( x - 2 = 0 \)
  • Solve for \( x \): \( x = 2 \)
This tells us that at \( x = 2 \), the function approaches but never actually reaches or equals any number. This is why you'll notice the curve of the graph getting infinitely close to the line \( x = 2 \). Think of a vertical asymptote as a boundary line for the graph in the horizontal direction.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a graph as the input values \( x \) become very large or very small. In our function \( y = \frac{1}{x-2} \), as \( x \) approaches infinity or negative infinity, the value \( 1/(x-2) \) gets closer and closer to zero:
  • As \( x \to \infty \), \( y \to 0 \)
  • As \( x \to -\infty \), \( y \to 0 \)
Therefore, \( y = 0 \) is the horizontal asymptote. This means the curve of the graph will get infinitely close to the x-axis but never actually touch it. When plotting your rational function graph, you'll see it leveling out along this horizontal line on both sides.
Undefined Points
Undefined points in a rational function occur where the function's denominator equals zero, resulting in a division by zero, which is mathematically undefined. In our specific rational function \( y = \frac{1}{x-2} \), the undefined point is at \( x = 2 \), since:
  • Setting the denominator to zero gives \( x - 2 = 0 \)
  • Solving this, we find \( x = 2 \)
At \( x = 2 \), the function doesn't have a value, meaning it cannot be graphed as it reaches towards infinity or negative infinity.
This is a key concept because recognizing undefined points helps you know where the function will have breaks or gaps, and where certain values don't exist on the graph. When you see an undefined point in a rational function, you'll often find a vertical asymptote at that spot.

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Most popular questions from this chapter

The table shows the average residential and transportation prices for energy consumption in the United States for the years \(2000-2008,\) as reported by the U.S. Department of Energy. The prices are given as dollars paid for one million BTU (British thermal units) of consumption. $$\begin{array}{lcc}\hline \text { Year } & \text { Residential (\$) } & \text { Transportation (\$) } \\\\\hline 2000 & 15 & 10 \\\2001 & 16 & 10 \\\2002 & 15 & 9 \\\2003 & 16 & 11 \\\2004 & 18 & 13 \\\2005 & 19 & 16 \\\2006 & 21 & 19 \\\2007 & 21 & 20 \\\2008 & 23 & 25 \\\\\hline\end{array}$$ a. Make a scatterplot of the data sets. b. Find and plot a regression line for each set of data points, and superimpose the lines on their scatterplots. c. What do you estimate as the average energy price for residential and transportation use for a million BTU in year \(2017 ?\) d. In looking at the trend lines, what do you conclude about the rising costs of energy across the two sectors of usage?

Find the function values. $$\sin ^{2} \frac{3 \pi}{8}$$

Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. \(y=1-x^{3},\) compressed horizontally by a factor of 3.

Graph the functions. $$y=1-x^{2 / 3}.$$

Graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in \(F: 8\) and applying an appropriate transformation. $$y=\sqrt{1-\frac{x}{2}}$$
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