Problem 36 - Algebra 2

Question

For Exercises \(34-37,\) use the following information. A hyperbola with asymptotes that are not perpendicular is called a nonrectangular hyperbola. Most of the hyperbolas you have studied so far are nonrectangular. A rectangular hyperbola is a hyperbola with perpendicular asymptotes. For example, the graph of \(x^{2}-y^{2}=1\) is a rectangular hyperbola. The graphs of equations of the form \(x y=c,\) where \(c\) is a constant, are rectangular hyperbolas with the coordinate axes as their asymptotes. Graph \(x y=-2\)

Step-by-Step Solution

Answer

The graph of \(xy = -2\) is a rectangular hyperbola with asymptotes \(x = 0\) and \(y = 0\), and branches in the second and fourth quadrants.

Step 1: Understand the Equation

The equation given is in the form of \(xy = c\), where \(c\) is a constant. Here, \(c = -2\). This form indicates a rectangular hyperbola because the product of the \(x\) and \(y\) coordinates is constant.

Step 2: Identify the Asymptotes

For the equation \(xy = c\), the asymptotes are the coordinate axes. This means the lines \(x = 0\) and \(y = 0\) are the asymptotes of the hyperbola. On the graph, the hyperbola approaches but never touches these lines.

Step 3: Determine the Branches of the Hyperbola

Since the hyperbola is symmetric about the origin, it has two branches located in opposite quadrants. For \(xy = -2\), the branches are located in the second (where \(x < 0, y > 0\)) and fourth quadrants (where \(x > 0, y < 0\)).

Step 4: Graph the Hyperbola

Start plotting points that satisfy \(xy = -2\). For example, if \(x = 1\), then \(y = -2\); if \(x = -2\), then \(y = 1\). Ensure these points are in the appropriate quadrants found in Step 3. Draw smooth curves through these points approaching the asymptotes without touching them.

Key Concepts

Asymptotes

Asymptotes play a crucial role in defining the shape and direction of hyperbolas. Imagine them as invisible boundaries that the hyperbola approaches but never quite reaches. The equation you might encounter, such as \(xy = c\), will often describe a rectangular hyperbola. For these, the asymptotes are the coordinate axes, i.e., the x-axis and y-axis.

• The x-axis (\(y = 0\)) is a horizontal line that the hyperbola will get indefinitely close to but never intersect in this context.
• The y-axis (\(x = 0\)) acts as the vertical boundary.

This characteristic is crucial when understanding how the graph forms. The hyperbola will not cross these axes, maintaining its distinctive open shape.

Coordinate Axes

The coordinate axes are the foundational lines in a two-dimensional graph, dividing the plane into four sections called quadrants. For a rectangular hyperbola like \(xy = c\), these axes do more than just guide the drawing.

• They serve as asymptotes, meaning the hyperbola approaches but never touches these lines.
• In the standard Cartesian coordinate system, the horizontal line is the x-axis, while the vertical line is the y-axis.

These axes essentially guide the path of the hyperbola, ensuring it has a recognizable shape and doesn't intersect with these crucial guiding lines.

Graphing Hyperbolas

To graph a hyperbola like \(xy = -2\), begin by identifying central points and understanding its symmetry.

• Start by selecting particular values for \(x\) and solving for \(y\). For example, if \(x = 1\), then \(y = -2\), and if \(x = -1\), then \(y = 2\). This gives you specific points on the graph.
• Ensure that these points fit the appropriate quadrants based on the relationship \(xy = -2\).

Graphing involves drawing smooth curves that connect these points in each of the quadrants, bearing in mind the hyperbola will never actually meet the coordinate axes—their asymptotes. The curves should sweep outwards, creating the recognizable hyperbolic shape.

Quadrants

The Cartesian plane is divided into four quadrants by the x and y coordinate axes. Each quadrant hosts different combinations of positive and negative values for \(x\) and \(y\), which is significant when mapping the path of a hyperbola.

• The first quadrant is for both positive \(x\) and \(y\) values.
• The second quadrant accommodates negative \(x\) but positive \(y\) values.
• The third has both \(x\) and \(y\) as negatives.
• The fourth is for positive \(x\) and negative \(y\) values.

For our equation \(xy = -2\), the branches will appear in the second and fourth quadrants. There, \(x\) and \(y\) have opposite signs, fulfilling the equation's requirement, \(xy < 0\). The hyperbola will be symmetrical around the origin, ensuring that its behavior mirrors across quadrants divided by these axes.