/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 574 Draw the graph of \(\mathrm{xy}=... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 574

Draw the graph of \(\mathrm{xy}=6\).

Short Answer

Expert verified
The graph of the equation \(xy=6\) is a hyperbola in Quadrants I and III, with the points \((-3, -2)\), \((-2, -3)\), \((-1, -6)\), \((1, 6)\), \((2, 3)\), and \((3, 2)\) plotted. The graph opens towards the positive \(x\) and \(y\) axes in Quadrant I and the negative \(x\) and \(y\) axes in Quadrant III.

Step by step solution

01

Rewrite the equation as a function of one variable

First, we need to rewrite the equation as a function of one variable so that we can easily plug in different values and find their corresponding points. If we rewrite the equation as a function of \(y\) in terms of \(x\), we'll have \(y(x) = \frac{6}{x}\). Notice that \(x\neq0\) because the denominator should not be zero.
02

Find some points for the graph

Now that we have the equation \(y(x) = \frac{6}{x}\), we can plug in different values for \(x\) to get their corresponding \(y\) values. Let's calculate the \(y\) values for \(x = -3\), \(x = -2\), \(x = -1\), \(x = 1\), \(x = 2\), and \(x = 3\). Here are the resulting points: 1. For \(x = -3\), \(y(-3) = \frac{6}{-3} = -2 \Rightarrow (-3, -2)\) 2. For \(x = -2\), \(y(-2) = \frac{6}{-2} = -3 \Rightarrow (-2, -3)\) 3. For \(x = -1\), \(y(-1) = \frac{6}{-1} = -6 \Rightarrow (-1, -6)\) 4. For \(x = 1\), \(y(1) = \frac{6}{1} = 6 \Rightarrow (1, 6)\) 5. For \(x = 2\), \(y(2) = \frac{6}{2} = 3 \Rightarrow (2, 3)\) 6. For \(x = 3\), \(y(3) = \frac{6}{3} = 2 \Rightarrow (3, 2)\)
03

Plot the points on the Cartesian plane

Now that we have the points for the graph, we can plot them on the Cartesian plane. Arrange the points on the plane as follows: 1. Point 1: \((-3, -2)\), located in Quadrant III 2. Point 2: \((-2, -3)\), located in Quadrant III 3. Point 3: \((-1, -6)\), located in Quadrant III 4. Point 4: \((1, 6)\), located in Quadrant I 5. Point 5: \((2, 3)\), located in Quadrant I 6. Point 6: \((3, 2)\), located in Quadrant I
04

Connect the points

Now that all the points are plotted, we can connect them to form the graph of the equation \(\mathrm{xy}=6\). The graph will be a hyperbola in Quadrants I and III, with its two branches opening towards the positive \(x\) and \(y\) axes in Quadrant I and the negative \(x\) and \(y\) axes in Quadrant III. The resulting graph should look like this: [INSERT GRAPH SHOWING HYPERBOLA IN QUADRANTS I AND III]

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