Problem 34 - Intermediate Algebra

Question

Are the graphs of \(x^{2}-y^{2}=0\) and \(y^{2}-x^{2}=0\) identical? Are the graphs of \(x^{2}-y^{2}=4\) and \(y^{2}-x^{2}=4\) identical? Explain your answers.

Step-by-Step Solution

Answer

The graphs of both \(x^2-y^2=0\) and \(y^2-x^2=0\) are identical, but the graphs of \(x^2-y^2=4\) and \(y^2-x^2=4\) are not identical.

Step 1: Rewrite the Equations

Start by rewriting each equation to see their forms more clearly. The equation \(x^2 - y^2 = 0\) can be rewritten as \(x^2 = y^2\). The equation \(y^2 - x^2 = 0\) can also be rewritten as \(y^2 = x^2\). For the second pair, \(x^2 - y^2 = 4\) remains as it is, and \(y^2 - x^2 = 4\) remains as it is.

Step 2: Analyze Equation Forms

Examine what these rewritten equations represent. Both \(x^2 = y^2\) and \(y^2 = x^2\) describe the same set of lines: \(x = y\) and \(x = -y\). Therefore, these are identical because they represent the same lines in the coordinate plane.

Step 3: Analyze Second Pair of Equations

For the second pair, \(x^2 - y^2 = 4\) describes a hyperbola with an opening along the x-axis, whereas \(y^2 - x^2 = 4\) describes a hyperbola with an opening along the y-axis. These are not identical as they have different orientations in the plane.

Key Concepts

Hyperbola Orientation

Hyperbolas can have different orientations depending on the terms of their equations. Determining the orientation is crucial to understanding their graphs on the coordinate plane. Consider the two equations given in the problem:

• \(x^2 - y^2 = 4\)
• \(y^2 - x^2 = 4\)

The orientation is determined by which variable has the positive leading coefficient.- For \(x^2 - y^2 = 4\), the positive \(x^2\) term shows that the hyperbola is oriented along the x-axis.- In contrast, \(y^2 - x^2 = 4\) has a positive \(y^2\) term, indicating that the hyperbola opens along the y-axis.This difference in orientation means these hyperbolas are not identical. They mirror each other, thus appearing rotated compared to one another on a graph.

Graphing Equations

Graphing hyperbolic equations involves identifying their standard form and plotting their shape correctly. A standard hyperbola equation form is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] for hyperbolas opening along the x-axis, and \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] for those along the y-axis.For the given equations:

• \(x^2 - y^2 = 4\) can be rewritten in standard form as \(\frac{x^2}{4} - \frac{y^2}{4} = 1\), showing it opens horizontally.
• \(y^2 - x^2 = 4\) becomes \(\frac{y^2}{4} - \frac{x^2}{4} = 1\), indicating it opens vertically.

To graph these hyperbolas:- First, rearrange the equation into its standard form.- Identify the center, vertices, and asymptotes from the equations.- Use the set opening direction to draw the curves accurately. Notice how they intersect the axes and plot these points clearly.

Coordinate Plane Analysis

Analyzing equations within the coordinate plane involves understanding both equations' visual implications. Looking at

• \(x^2 - y^2 = 0\)
• \(y^2 - x^2 = 0\)

Rewriting these implies they both transform to \(x = \pm y\). - They represent the same pair of intersecting lines in the coordinate plane.- This means the graphs of these equations are identical as they express the same geometric shape.Analyzing hyperbolas such as \(x^2 - y^2 = 4\) and \(y^2 - x^2 = 4\) requires understanding their interaction with the coordinate plane:- Recognize that the change in sign between terms forms hyperbolas that face perpendicular directions.- This orientation means they intersect differently on the plane, emphasizing visualization's importance in seeing these differences.