Problem 8 - 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. $$ 2 y^{2} x-4 x^{2}=x y^{4} $$

Step-by-Step Solution

Answer

The graph is symmetric about the x-axis and the origin. It has the x-intercept (0,0) and every point (0,y) as y-intercepts.

Step 1: Find the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \). Substitute \( y = 0 \) into the given equation: \( 2 \cdot 0^2 \cdot x - 4x^2 = x \cdot 0^4 \). This simplifies to \( -4x^2 = 0 \). Solving this equation gives \( x = 0 \). Thus, the x-intercept is at the origin \((0, 0)\).

Step 2: Find the y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation and solve for \( y \). Substitute \( x = 0 \) into the equation: \( 2y^2 \cdot 0 - 4 \cdot 0 = 0 \cdot y^4 \), which holds true for any value of \( y \). Hence, every point of the form \((0, y)\) is a y-intercept. Therefore, the entire y-axis is composed of y-intercepts.

Step 3: Simplify the Original Equation

To better analyze symmetry, let's rearrange and factor the given equation: \( 2y^{2}x - xy^{4} = 4x^{2} \). Factor out common terms: \( x(2y^{2} - y^{4}) = 4x^{2} \). Dividing both sides by \( x \) (assuming \( x eq 0 \)): \( 2y^{2} - y^{4} = 4x \).

Step 4: Test for Symmetry about the x-axis

Replace \( y \) with \(-y \) in the equation: \( 2(-y)^{2}x - 4x^{2} = x (-y)^{4} \). Simplifying, we get \( 2y^2x - 4x^{2} = xy^{4} \), which is the same as the original equation. Therefore, the graph is symmetric about the x-axis.

Step 5: Test for Symmetry about the y-axis

Replace \( x \) with \(-x \) in the equation: \( 2y^{2}(-x) - 4(-x)^{2} = -x y^{4} \). This simplifies to \(-2y^2x - 4x^{2} = -xy^4\), which simplifies to the negative of the original equation. Thus, the graph is not symmetric about the y-axis.

Step 6: Test for Symmetry about the Origin

Replace both \( x \) with \(-x \) and \( y \) with \(-y \) in the equation: \( 2(-y)^2(-x) - 4(-x)^2 = -x(-y)^4 \), simplifying to \(-2y^2x - 4x^2 = -xy^4\), which becomes \(2y^2x - 4x^2 = xy^4\), the original equation. Thus, the graph is symmetric about the origin.

Key Concepts

x-intercepts

The x-intercepts of a graph are points where the graph crosses the x-axis. To find them, you set the y variable to zero in the equation and solve for x.
In the given equation, substituting \( y = 0 \) returns: \( 2 \times 0^2 \times x - 4x^2 = x \times 0^4 \). Simplified, this leads to \( -4x^2 = 0 \). The solution \( x = 0 \) indicates an x-intercept at the origin. Thus, the graph has an x-intercept at \( (0, 0) \).
This means the graph only touches the x-axis at the origin point.

y-intercepts

The y-intercepts of a graph occur where the graph meets the y-axis. To discover these, you set x to zero in your equation and resolve for y.
With the current equation, substituting \( x = 0 \) yields: \( 2y^2 \times 0 - 4 \times 0 = 0 \times y^4 \). This remains true across a spectrum of y-values. Consequently, every point along the y-axis, denoted \( (0, y) \), serves as a y-intercept.
Therefore, the entire y-axis acts as a line of y-intercepts, emphasizing how extensively the graph interacts with the y-axis.

equation symmetry

Equation symmetry in a graph determines if the graph appears unchanged following certain transformations.

**Symmetry about the x-axis**:
Substitute y with \(-y\) in the equation to check for symmetry about the x-axis. The calculation: \( 2(-y)^2 x - 4x^2 = x (-y)^4 \) equates to the initial equation, confirming symmetry about the x-axis.

**Symmetry about the y-axis**:
Replace x with \(-x\) in the equation. The equation becomes \( -2y^2 x - 4x^2 = -xy^4 \), yielding a result that is not the same as the original, showing no symmetry about the y-axis.

**Symmetry about the origin**:
For origin symmetry, switch both x and y with \(-x\) and \(-y\) respectively: \( 2 (-y)^2(-x) - 4(-x)^2 = -x (-y)^4 \), resulting in the original equation form. This confirms symmetry about the origin, meaning the graph looks identical on each side when rotated 180 degrees.

graph analysis

Conducting a graph analysis allows you to gather insights into its behavior and features.
By identifying intercepts and understanding symmetry, you can map out the graph's nature.

• **X-intercepts**: Determines where the graph crosses or touches the x-axis, showing points of zero y-value.
• **Y-intercepts**: Reveals where the graph intersects the y-axis, highlighting the passage across zero x-values.
• **Symmetry Analysis**: Understanding symmetry simplifies predicting a graph's complete behavior and understanding its balance about axes or the origin.

Ultimately, such analysis equips you with a firm understanding of how equations transform into graphical representations, revealing a broader context beyond mere equations.