Question

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

Short answer

To draw the graph of the equation \(xy = 6\) (or \(y = \frac{6}{x}\)), we create a table of values (such as (-3,-2), (-2,-3), (-1,-6), (1,6), (2,3), (3,2)). Plot these points on a graph, and then sketch a smooth curve connecting them. The resulting graph represents the equation \(xy = 6\).

Step-by-step solution

Rewrite the equation in the form of y = f(x)

To rewrite the given equation xy = 6 in the form of y = f(x), we will simply divide both sides of the equation by x: \[y = \frac{6}{x}\]

Create a table of values

Now that we have the equation in the form y = f(x), let's create a table of values with some chosen values for x and the corresponding values for y: x | y -3 | -2 -2 | -3 -1 | -6 1 | 6 2 | 3 3 | 2 These are some of the points that lie on the curve of the equation y = 6/x.

Plot the points on the graph

Using the table of values, plot the points on a graph paper or graphing software: (-3,-2), (-2,-3), (-1,-6), (1,6), (2,3), (3,2)

Sketch the curve using the plotted points as a guide

Now that we have plotted some representative points for the equation y = 6/x, we can see the general shape of the curve. Sketch the curve by connecting the points smoothly and extending the curve in both directions. With these steps, the graph of the equation xy = 6 (or y = 6/x) has been successfully drawn.

Key concepts

Understanding Asymptotes

When we talk about rational functions like \( y = \frac{6}{x} \), asymptotes are a vital concept to grasp. An asymptote is a line that the graph of a function approaches but never actually touches. For this equation, we mainly work with two types of asymptotes:

• Vertical Asymptote: This occurs when the denominator of a rational function is zero. For \( y = \frac{6}{x} \), the denominator becomes zero when \( x = 0 \). Therefore, the line \( x = 0 \) is a vertical asymptote. The graph will get infinitely close to this line but never cross it.
• Horizontal Asymptote: This is determined by the behavior of the graph as \( x \) gets very large or very small. In the case of \( y = \frac{6}{x} \), as \( x \) approaches infinity or negative infinity, \( y \) approaches zero. Thus, \( y = 0 \) is a horizontal asymptote.

Asymptotes play a crucial role as they define the behavior of the graph at extreme values. Understanding them helps in drawing more accurate graphs.

Exploring the XY-Plane

The XY-plane, also known as the Cartesian coordinate system, is where we plot the graph of equations like \( y = \frac{6}{x} \). It's a two-dimensional plane consisting of a horizontal axis (X-axis) and a vertical axis (Y-axis).
When plotting points, each has an \( (x, y) \) coordinate:

• X-Values: Determine how far left or right the point is from the origin (0,0).
• Y-Values: Determine how far up or down the point is from the origin.

For the equation \( y = 6/x \), we chose points such as \((-3, -2)\) and \((1, 6)\). By plotting these on the XY-plane and connecting them, the characteristic hyperbolic shape of the graph is formed.
Grasping the XY-plane is essential for visualizing and understanding different types of functions and their behaviors.

Inverse Variation and Its Impact

Inverse variation is a key concept in equations like \( y = \frac{6}{x} \), representing a relationship where one variable increases while the other decreases. Simply put, as \( x \) gets larger, \( y \) becomes smaller, and vice versa. This is unlike direct variation where both variables increase or decrease together.
For example, when \( x = 1 \), \( y = 6 \), and as \( x \) increases to \( 3 \), \( y \) decreases to \( 2 \). This inverse behavior describes the equation's dynamics, creating the distinctive hyperbolic graph.
Inverse variation is seen in many real-life situations, such as speed and time for a constant distance. Understanding this concept helps in solving various mathematical and practical problems.