Problem 7 - 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. $$ x^{2} y^{4}-2 x^{4}=1 $$

Step-by-Step Solution

Answer

No intercepts; symmetric with respect to the x-axis, y-axis, and origin.

Step 1: Find the y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation \( x^2 y^4 - 2x^4 = 1 \). This simplifies to \( 0 - 0 = 1 \), which is not possible. Therefore, there are no y-intercepts.

Step 2: Find the x-intercepts

For x-intercepts, set \( y = 0 \). Substitute into the equation: \( x^2(0)^4 - 2x^4 = 1 \), which results in \( -2x^4 = 1 \). Solving gives \( x^4 = -\frac{1}{2} \), which has no real solution since a fourth power cannot be negative. Hence, there are no x-intercepts either.

Step 3: Check symmetry with respect to the x-axis

For x-axis symmetry, replace \( y \) with \( -y \). The equation becomes \( x^2(-y)^4 - 2x^4 = 1 \), simplifying to \( x^2y^4 - 2x^4 = 1 \), which matches the original equation. Therefore, the graph is symmetric with respect to the x-axis.

Step 4: Check symmetry with respect to the y-axis

For y-axis symmetry, replace \( x \) with \( -x \). The equation becomes \( (-x)^2y^4 - 2(-x)^4 = 1 \), simplifying to \( x^2y^4 - 2x^4 = 1 \), matching the original equation. Therefore, the graph is symmetric with respect to the y-axis.

Step 5: Check symmetry with respect to the origin

For origin symmetry, replace \( x \) with \( -x \) and \( y \) with \( -y \). The equation becomes \( (-x)^2(-y)^4 - 2(-x)^4 = 1 \), simplifying to \( x^2y^4 - 2x^4 = 1 \), which matches the original equation. Therefore, the graph is symmetric with respect to the origin.

Key Concepts

Intercepts

Intercepts are the points where a graph crosses the axes. Finding these points helps us understand more about the graph's behavior and position. For this particular problem, we're analyzing the graph of the equation \( x^2 y^4 - 2x^4 = 1 \). Y-Intercepts: To find y-intercepts, we set \( x = 0 \) and solve. Substituting, we get \( 0 - 0 = 1 \), which isn't possible. This means no y-intercepts exist. The graph never crosses the y-axis.
X-Intercepts: For x-intercepts, replace \( y \) with 0. The equation becomes \( -2x^4 = 1 \), leading to \( x^4 = -\frac{1}{2} \). Since a fourth power cannot be negative, there are no x-intercepts. The graph doesn't cross the x-axis either.Understanding intercepts ensures we know where and if the graph touches or crosses the axes.

Fourth Power Functions

Fourth power functions include terms where variables are raised to the fourth power like \( x^4 \) or \( y^4 \). These functions have specific characteristics that can affect the graph's shape and positions.

• Even Powers: The expression \( x^4 \) is always non-negative, meaning it yields either a positive value or zero.
• Symmetry: Fourth power functions can contribute to symmetry since even powers behave similarly in positive and negative directions.

An equation involving \( x^4 \) or \( y^4 \) suggests symmetry, as in our given equation. Even when the sum or difference includes such terms, they often reinforce the graph's symmetry along certain axes.

Symmetry

Graph symmetry simplifies the analysis of graphs by reducing the complexity of their shapes. Assessing symmetry involves checking if changing the sign of variables keeps the equation identical. Types of Symmetry:

• X-axis Symmetry: Substitute \( y \) with \( -y \). The change doesn't alter the original equation, \( x^2y^4 - 2x^4 = 1 \). Thus, the graph is symmetric about the x-axis.
• Y-axis Symmetry: Substitute \( x \) with \( -x \). The equation remains \( x^2y^4 - 2x^4 = 1 \), verifying symmetry about the y-axis.
• Origin Symmetry: Substitute both \( x \) and \( y \) with \( -x \) and \( -y \) respectively. Again, the equation doesn't change. This confirms symmetry about the origin.

All types of symmetry in this case lead to a well-balanced graph, exhibiting symmetry regarding both axes and the origin.

Real Solutions

Real solutions are essential when solving equations involving polynomials or functions, representing values where the equation holds true. In this context, real solutions determine whether intercepts exist. For fourth power equations like \( x^4 \), consider these points:

• Positive Results: Fourth powers can return positive numbers or zero but never negative values.
• No Real Solutions: If an equation like \( x^4 = -\frac{1}{2} \) arises, it's crucial to recognize its impossibility in real numbers. No real value for \( x \) exists that satisfies this condition.

Understanding and identifying when real solutions don't exist is critical in assessing the full behavior of the graph. It prevents incorrect assumptions regarding intercepts or other features.