Problem 6 - 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. $$ x^{4}=3 y^{3}+4 $$

Step-by-Step Solution

Answer

X-intercepts: \((\sqrt{2}, 0), (-\sqrt{2}, 0)\); Y-intercept: \((0, -\sqrt[3]{4/3})\). The graph is symmetric with respect to the y-axis.

Step 1: Determine the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \). The equation becomes \( x^4 = 3(0)^3 + 4 \), which simplifies to \( x^4 = 4 \). Solving \( x^4 = 4 \) gives \( x = \pm \sqrt[4]{4} \), which are \( x = \pm \sqrt{2} \). Therefore, the x-intercepts are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

Step 2: Determine the y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation and solve for \( y \). The equation becomes \( 0^4 = 3y^3 + 4 \), which simplifies to \( 0 = 3y^3 + 4 \). Solving \( 0 = 3y^3 + 4 \) gives \( y^3 = -\frac{4}{3} \), so \( y = -\sqrt[3]{\frac{4}{3}} \). Thus, the y-intercept is \( (0, -\sqrt[3]{\frac{4}{3}}) \).

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

Replace \( y \) with \( -y \) to test for symmetry with respect to the x-axis. The equation becomes \( x^4 = 3(-y)^3 + 4 \), which simplifies to \( x^4 = -3y^3 + 4 \). Since this is not equivalent to the original equation \( x^4 = 3y^3 + 4 \), the graph is not symmetric with respect to the x-axis.

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

Replace \( x \) with \( -x \) to test for symmetry with respect to the y-axis. The equation becomes \( (-x)^4 = 3y^3 + 4 \), which simplifies to \( x^4 = 3y^3 + 4 \). Since this equation is equivalent to the original equation, the graph is symmetric with respect to the y-axis.

Step 5: Test for symmetry with respect to the origin

Replace \( x \) with \( -x \) and \( y \) with \( -y \) to test for symmetry with respect to the origin. The equation becomes \( (-x)^4 = 3(-y)^3 + 4 \), which simplifies to \( x^4 = -3y^3 + 4 \). Since this is not equivalent to the original equation \( x^4 = 3y^3 + 4 \), the graph is not symmetric with respect to the origin.

Key Concepts

Intercepts

Intercepts serve as the points at which a graph crosses the axes in a Cartesian coordinate system. Understanding intercepts is crucial because they help us visually interpret equations. There are two main types of intercepts: the x-intercept and the y-intercept. Each provides valuable insights into the behavior of an equation's graph. Let's dive deeper into what each type of intercept represents and how to calculate them.

X-Intercept

The x-intercepts of a graph are the points where the graph crosses the x-axis. To find the x-intercepts of an equation, substitute 0 for the variable y and solve for x. In the given equation, setting \( y = 0 \) transforms it into \( x^4 = 4 \). Solving this, we find that \( x = \pm \sqrt{2} \). Thus, the graph intersects the x-axis at the points \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \). These intercepts are essential in sketching the graph as they provide exact points of intersection with the x-axis.

Y-Intercept

The y-intercept of a graph is the point where it crosses the y-axis. To find the y-intercept, you set x to zero and solve for y. In our example, by substituting \( x = 0 \) into the equation, we simplify it to \( 0 = 3y^3 + 4 \). Solving this for y yields \( y = -\sqrt[3]{\frac{4}{3}} \). Therefore, the y-intercept occurs at \( (0, -\sqrt[3]{\frac{4}{3}}) \). This point helps us understand where the graph intersects the y-axis, which aids in constructing an accurate representation of the graph's behavior.

Symmetry Tests

Symmetry tests help us understand the visual balance of a graph concerning particular axes or the origin. We have three common types of symmetry checks: with respect to the x-axis, the y-axis, and the origin.

• X-axis Symmetry: Replace y with -y in the equation. If the resulting equation is identical to the original, the graph is symmetric about the x-axis. For our equation, this is not the case.
• Y-axis Symmetry: Replace x with -x in the equation. If the modified equation is the same as the original, the graph is symmetric with respect to the y-axis. Here, this condition is satisfied.
• Origin Symmetry: Replace both x with -x and y with -y. If the equation remains unchanged, the graph is symmetric about the origin. The given equation does not fulfill this condition.

Understanding these symmetry properties simplifies graph sketching and helps in identifying unique graph characteristics.