Problem 31 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x y=2 $$

Step-by-Step Solution

Answer

The hyperbola \(xy=2\) has no x- or y-intercepts and is symmetric about the origin.

Step 1: Understand the Equation

The equation given is a product of two variables: \(xy = 2\). This is a hyperbola equation, meaning it will produce a curve when graphed.

Step 2: Make a Table of Values

To make a table of values, choose various values for either \(x\) or \(y\) and solve for the other variable. For example:- If \(x = 1\), then \(y = \frac{2}{1} = 2\).- If \(x = 2\), then \(y = \frac{2}{2} = 1\).- If \(x = -1\), then \(y = \frac{2}{-1} = -2\).- If \(x = 0.5\), then \(y = \frac{2}{0.5} = 4\).

Step 3: Sketch the Graph

Using the table of values, plot the corresponding points on a Cartesian plane:- \((1, 2)\)- \((2, 1)\)- \((-1, -2)\)- \((0.5, 4)\)Since this is a hyperbola, the graph will consist of two separate branches.

Step 4: Find the X-Intercepts

To find x-intercepts, set \(y = 0\) in the equation \(xy = 2\): \[x \times 0 = 2\]. This does not yield a valid intercept since any number multiplied by zero is zero, and 2 cannot be zero. Therefore, there are no x-intercepts.

Step 5: Find the Y-Intercepts

To find y-intercepts, set \(x = 0\) in the equation \(xy = 2\):\[0 \times y = 2\].This also does not yield a valid intercept for the same reason as above; there are no y-intercepts.

Step 6: Test for Symmetry

Symmetry can be around the x-axis, y-axis, or origin. For the origin symmetry test, replace \(x\) and \(y\) with \(-x\) and \(-y\):\((-x)(-y) = 2\),which simplifies to \(xy = 2\), showing the equation is symmetric about the origin.

Key Concepts

Table of Values for Hyperbola

A table of values is essential for graphing an equation, as it helps identify specific coordinate points to plot. For the equation \(xy = 2\), you can choose values for \(x\) or \(y\) and solve for the other variable. By doing this for a range of values, you build a set of coordinates that reflect the relationship in the equation.

• For \(x = 1\), \(y\) is \(2\).
• For \(x = 2\), \(y\) is \(1\).
• For \(x = -1\), \(y\) is \(-2\).
• For \(x = 0.5\), \(y\) is \(4\).

These points: \((1, 2), (2, 1), (-1, -2), (0.5, 4)\), guide the sketch of the hyperbola on a Cartesian plane. Each point is a solution to the equation and represents a key part of the hyperbola's shape.

Understanding X-Intercepts

The x-intercepts of a graph are where the curve crosses the x-axis, meaning these points have a \(y\)-value of zero. To find them, we set \(y = 0\) in the equation \(xy = 2\):\[x \times 0 = 2\]This equation cannot be solved for a real number because any number multiplied by zero results in zero, not two. As a result, the graph of this particular hyperbola does not have x-intercepts. No points exist where the graph crosses the x-axis, aligning with the fact that the hyperbola is open and does not meet the horizontal axis.

Locating Y-Intercepts

Y-intercepts occur where the graph crosses the y-axis, so the \(x\)-value needs to be zero. To find them, set \(x = 0\) in the same equation:\[0 \times y = 2\]Again, as with the x-intercepts, this equation does not yield a solution. With zero on the left, it cannot equal two, indicating that this hyperbola does not cross the y-axis. Hence, there are no y-intercepts for this equation. This is characteristic of hyperbolas where neither x nor y can independently equate to zero without breaking the relationship defined by the equation.

Checking for Symmetry

Symmetry in a graph means that one part of the graph is a mirror image of another. A hyperbola like \(xy = 2\) can have several types of symmetry. To test for symmetry about the origin, replace \(x\) and \(y\) with \(-x\) and \(-y\):\[(-x)(-y) = 2\]Simplifying this, we get \(xy = 2\), implying the graph is symmetric about the origin. This means the graph can be rotated 180 degrees about the origin and it will look the same. Understanding symmetry helps in predicting the shape and the behavior of the hyperbola without having to plot countless points.