Problem 16 - Calculus with Concepts in Calculus

Question

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ y^{2}=\frac{x^{2}+1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The graph has no real intercepts and is symmetric with respect to the x-axis, y-axis, and origin.

Step 1: Find the x-intercepts

To find the x-intercepts, we set \(y = 0\) in the equation and solve for \(x\). Substituting \(y = 0\) gives: \(0^2 = \frac{x^2 + 1}{x^2 - 1}\). Simplifying, we get \(0 = \frac{x^2 + 1}{x^2 - 1}\), which implies the fraction is undefined where \(x^2 - 1 = 0\), giving no real \(x\)-intercepts.

Step 2: Find the y-intercepts

To find the y-intercepts, we set \(x = 0\) in the equation and solve for \(y\). Substituting \(x = 0\) gives: \(y^2 = \frac{0^2 + 1}{0^2 - 1}\). So \(y^2 = -1\), which has no real solutions, implying there are no real \(y\)-intercepts.

Step 3: Test for symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, check if replacing \(y\) with \(-y\) leaves the equation unchanged. Substitute \(-y\) for \(y\) in the equation: \((-y)^2 = \frac{x^2 + 1}{x^2 - 1}\), which simplifies back to \(y^2 = \frac{x^2 + 1}{x^2 - 1}\). Therefore, the graph is symmetric with respect to the x-axis.

Step 4: Test for symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, check if replacing \(x\) with \(-x\) leaves the equation unchanged. Substitute \(-x\) for \(x\): \(y^2 = \frac{(-x)^2 + 1}{(-x)^2 - 1}\). This simplifies back to \(y^2 = \frac{x^2 + 1}{x^2 - 1}\). Thus, the graph is symmetric with respect to the y-axis.

Step 5: Test for symmetry with respect to the origin

To test for symmetry with respect to the origin, check if replacing both \(x\) with \(-x\) and \(y\) with \(-y\) leaves the equation unchanged. Substitute \(-x\) for \(x\) and \(-y\) for \(y\): \((-y)^2 = \frac{(-x)^2 + 1}{(-x)^2 - 1}\). This simplifies back to \(y^2 = \frac{x^2 + 1}{x^2 - 1}\). Therefore, the graph is also symmetric with respect to the origin.

Key Concepts

Graph Intercepts

Graph intercepts are points where a graph crosses the axes. These points include x-intercepts, where the graph crosses the x-axis, and y-intercepts, where it crosses the y-axis. To identify these intercepts, substitute zero into the other variable and solve for the remaining one. For our specific equation defined by the formula \( y^2 = \frac{x^2 + 1}{x^2 - 1} \), we start by analyzing potential intercepts.

• X-Intercepts: Set \( y = 0 \). This turns the equation into \( 0 = \frac{x^2 + 1}{x^2 - 1} \). Solving, we find no values make this fraction zero, implying no real x-intercepts.
• Y-Intercepts: Set \( x = 0 \). This results in \( y^2 = \frac{1}{-1} \), or \( y^2 = -1 \), which has no real solutions, indicating no real y-intercepts.

Understanding intercepts offers insight into the behavior of the graph with respect to the axes.

Symmetry with Respect to Axis

Symmetry in a graph determines its visual balance around certain lines or points. In this context, it can refer to symmetry with respect to the x-axis, y-axis, or the origin. Each kind of symmetry involves specific substitutions in the equation:

• X-axis Symmetry: Replace \( y \) with \( -y \). For our equation, substituting \( -y \) yields the same expression. Hence, symmetry with the x-axis is confirmed.


• Y-axis Symmetry: Replace \( x \) with \( -x \). Our equation remains unchanged, so it is symmetric with respect to the y-axis.


• Origin Symmetry: Replace both \( x \) with \( -x \) and \( y \) with \( -y \). Again, the equation stays the same, indicating symmetry with respect to the origin.

These symmetries suggest that our function is visually balanced across both axes and the origin.

Equation Analysis

Analyzing equations involves understanding the structure and characteristics of mathematical expressions. For the equation \( y^2 = \frac{x^2 + 1}{x^2 - 1} \), several aspects can be explored:

• Rational Expression Structure: The equation consists of a rational expression with a quadratic numerator and denominator. The fractions suggest possible vertical asymptotes when the denominator equals zero. Here, \( x^2 - 1 = 0 \) occurs at \( x = \pm 1 \), indicating asymptotes at these values.


• Domain Considerations: The expression being undefined at \( x = \pm 1 \) means these values are not in the domain. Specifically, \( x \) cannot equal \( \pm 1 \) without causing undefined operations.


• Absence of Real Intercepts: As previously analyzed, the equation has no real x or y-intercepts, shaping the way we perceive the graph's interaction with the axes.

This equation analysis helps to further understand the limitations and characteristics of the graph, guiding us on how the graph behaves across the coordinate plane.