Problem 5 - 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. $$ x^{4}=3 y^{3} $$

Step-by-Step Solution

Answer

Intercepts: \((0,0)\). Symmetric with respect to the y-axis.

Step 1: Finding the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation \( x^4 = 3y^3 \). This simplifies to \( x^4 = 0 \), resulting in \( x = 0 \) as the only x-intercept.

Step 2: Finding the y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation \( x^4 = 3y^3 \). This simplifies to \( 0 = 3y^3 \), resulting in \( y = 0 \) as the only y-intercept.

Step 3: Deciding on x-axis symmetry

For symmetry with respect to the x-axis, replace \( y \) with \( -y \) in the equation. This gives \( x^4 = 3(-y)^3 = -3y^3 \), which is not equivalent to the original equation, indicating no symmetry with respect to the x-axis.

Step 4: Deciding on y-axis symmetry

For symmetry with respect to the y-axis, replace \( x \) with \( -x \) in the equation. This results in \( (-x)^4 = 3y^3 \), which simplifies back to \( x^4 = 3y^3 \). Therefore, the graph is symmetric with respect to the y-axis.

Step 5: Deciding on origin symmetry

For symmetry with respect to the origin, replace \( x \) with \( -x \) and \( y \) with \( -y \). The equation becomes \( (-x)^4 = 3(-y)^3 = -3y^3 \), which is not equivalent to the original equation, indicating no symmetry with respect to the origin.

Key Concepts

Intercepts

An intercept on a graph is the point where the graph crosses either the x-axis or the y-axis. It's where either the x-coordinate or the y-coordinate is zero. Let's look at how we find both types of intercepts. To find x-intercepts, you set y equal to zero in the equation and solve for x. In our example, this involves setting the equation \( x^4 = 3y^3 \) to \( x^4 = 0 \) because \( y = 0 \). Solving \( x^4 = 0 \) gives you \( x = 0 \), which means the only x-intercept is at (0,0).
Similarly, to find the y-intercepts, you set x equal to zero and solve for y. Again, using our equation, we set it to \( 0 = 3y^3 \) because \( x = 0 \). Solving \( 3y^3 = 0 \) results in \( y = 0 \), indicating the only y-intercept is also at (0,0). In this case, both the x-intercept and y-intercept lie at the origin, (0,0).

X-Axis Symmetry

Graph symmetry with respect to the x-axis means that the graph looks the same above and below the x-axis. To check for this kind of symmetry, substitute \( y \) with \( -y \) and see if the equation remains unchanged. In our equation, changing \( y \) to \( -y \) results in \( x^4 = 3(-y)^3 \), which simplifies to \( x^4 = -3y^3 \). Since this is not equivalent to the original equation \( x^4 = 3y^3 \), we conclude that the graph does not exhibit x-axis symmetry. In simpler terms, flipping the graph over the x-axis wouldn't align perfectly with the original drawing.

Y-Axis Symmetry

Y-axis symmetry implies that the graph remains unchanged if reflected over the y-axis. This occurs when substituting \( x \) with \( -x \) in the equation still results in the same equation. Let's test this using our equation. By substituting \( -x \) for \( x \), the equation becomes \( (-x)^4 = 3y^3 \), which simplifies back to \( x^4 = 3y^3 \). This confirms that the equation remains unchanged, indicating that the graph is symmetric with respect to the y-axis. In this case, folding the graph along the y-axis will not change how it looks.

Origin Symmetry

Origin symmetry indicates that the graph looks identical when rotated 180 degrees around the origin. To test for origin symmetry, both \( x \) and \( y \) are replaced by \( -x \) and \( -y \), respectively. In our equation, replacing these values results in \( (-x)^4 = 3(-y)^3 \), which simplifies to \( x^4 = -3y^3 \). Since this equation is not equal to the original \( x^4 = 3y^3 \), the graph does not have origin symmetry. This lack of symmetry means that if you rotate the graph upside down around the origin, it doesn’t match up with the original position of the graph.