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Chapter 2: Problem 24

Sketch the graphs of the following functions for \(x > 0\). $$y=\frac{2}{x}$$

Short Answer

Expert verified
The graph has vertical asymptote at \(x=0\) and horizontal asymptote at \(y=0\). Points can be (1,2), (2,1), and (4,0.5).

Step by step solution

01

- Identify the function type

The given function is \(y = \frac{2}{x}\). This is a type of rational function where a constant is divided by the variable x.
02

- Determine key characteristics

For rational functions of the form \(y = \frac{c}{x}\), the graph will have a vertical asymptote (where the function is undefined) at \(x=0\), and a horizontal asymptote at \(y=0\). Since \(c=2\), the graph will be shaped similarly to \(y=\frac{1}{x}\), but steeper.
03

- Create a table of values

Choose values of x greater than 0 to compute corresponding y-values. For instance, \(x = 0.5, 1, 2, 4\). Calculate the corresponding y-values: \(y(0.5) = \frac{2}{0.5} = 4\), \(y(1) = \frac{2}{1} = 2\), \(y(2) = \frac{2}{2} = 1\), \(y(4) = \frac{2}{4} = 0.5\). This gives points (0.5, 4), (1, 2), (2, 1), and (4, 0.5).
04

- Plot the points

On a coordinate system, plot the points (0.5, 4), (1, 2), (2, 1), and (4, 0.5). Extending these points will show the shape of the curve.
05

- Draw the asymptotes

Draw a dashed line at \(x=0\) to represent the vertical asymptote and at \(y=0\) for the horizontal asymptote.
06

- Sketch the graph

Using the plotted points and the asymptotes, sketch the hyperbolic curve, making sure it approaches the asymptotes but never touches them.

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Key Concepts

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

Vertical asymptote
A vertical asymptote is a line that the graph of a function approaches but never actually touches. It occurs in rational functions when the denominator is equal to zero. For the given function \[ y = \frac{2}{x} \], the denominator is x. Therefore, the vertical asymptote is at \[ x = 0 \]. This means that as x gets closer to 0 from either the positive or negative side, the value of y increases or decreases without bounds. To indicate a vertical asymptote on a graph, draw a dashed line where the function is undefined, which in this case is the y-axis (x = 0). Remember, the function will approach this line infinitely but will never intersect it.
Horizontal Asymptote
A horizontal asymptote is a line that the graph of the function approaches as x tends to infinity or negative infinity. For the function \[ y = \frac{2}{x} \], observe what happens as the absolute value of x becomes very large. The value of y gets closer and closer to 0. This is because the numerator (2 in this case) remains constant while the denominator (x) grows. Consequently, the fraction as a whole diminishes. The equation for the horizontal asymptote is thus \[ y = 0 \]. To represent this on the graph, draw a dashed line along the x-axis. Just like with vertical asymptotes, the function will get infinitely close to this line without crossing it.
Rational Function
A rational function is a fraction in which the numerator and the denominator are both polynomials. The simplest example of a rational function is \[ y = \frac{2}{x} \]. This function highlights some key characteristics of rational functions: asymptotes and behavior near extremes. Use these steps to analyze any rational function:
  • Identify the form: For \[ y = \frac{2}{x} \], note the constant numerator and variable denominator.
  • Find the vertical asymptote: Set the denominator to zero and solve. Here, x = 0.
  • Identify the horizontal asymptote: Check the behavior as x approaches infinity. For \[ y = \frac{2}{x} \], it is y = 0.
  • Calculate key points: Plug in various x values to understand the function's behavior and manually plot those points. This helps visualize any drastic changes or trends in the graph.
Through these steps, understanding and graphing a rational function becomes a systematic and straightforward process. Pay close attention to the presence and interaction of asymptotes to accurately depict the graph's behavior.

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Most popular questions from this chapter

Find two positive numbers, \(x\) and \(y,\) whose product is 100 and whose sum is as small as possible.

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The demand equation for a company is \(p=200-3 x,\) and the cost function is $$Q(x)=75+80 x-x^{2}, \quad 0 \leq x \leq 40.$$ (a) Determine the value of \(x\) and the corresponding price that maximize the profit. (b) If the government imposes a tax on the company of \(\$ 4\) per unit quantity produced, determine the new price that maximizes the profit. (c) The government imposes a tax of \(T\) dollars per unit quantity produced (where \(0 \leq T \leq 120\) ), so the new cost function is $$C(x)=75+(80+T) x-x^{2}, \quad 0 \leq x \leq 40.$$ Determine the new value of \(x\) that maximizes the company's profit as a function of \(T .\) Assuming that the company cuts back production to this level, express the tax revenues received by the government as a function of \(T\). Finally, determine the value of \(T\) that will maximize the tax revenue received by the government.
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