Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{4}+y^{4}=16 $$

Step-by-Step Solution

Answer

The graph is a symmetrical curve extending to (±2, 0) and (0, ±2).

Step 1: Understanding the Equation

The given equation is \( x^4 + y^4 = 16 \). This is not a standard line or circle equation, but instead represents a type of curve known as a superellipse. The equation is symmetric in both variables since swapping \( x \) with \( y \) doesn't change the equation.

Step 2: Checking for Symmetries

The equation is symmetric about the y-axis and the x-axis. This means if \((x, y)\) is a solution, then \((-x, y)\), \((x, -y)\), and \((-x, -y)\) are also solutions. This helps us understand that the graph will be symmetric in all quadrants.

Step 3: Finding x-Intercepts

For x-intercepts, set \( y = 0 \): \( x^4 + 0^4 = 16 \) simplifies to \( x^4 = 16 \), giving \( x = 2 \) and \( x = -2 \). Therefore, the x-intercepts are \((2, 0)\) and \((-2, 0)\).

Step 4: Finding y-Intercepts

For y-intercepts, set \( x = 0 \): \( 0^4 + y^4 = 16 \) simplifies to \( y^4 = 16 \), giving \( y = 2 \) and \( y = -2 \). Therefore, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Step 5: Plotting the Graph

Sketch the graph based on the intercepts and symmetries. The shape is like a diamond with curves rather than straight lines. The boundaries extend to the intercepts. Carefully plot points by testing additional values to ensure the graph's curves are accurate, ensuring you showcase the symmetry found in all four quadrants.

Key Concepts

Symmetry in Graphs

Graphs can exhibit symmetry, making them easier to interpret and sketch. Symmetry can occur about the x-axis, y-axis, or origin. For the equation of a superellipse like \(x^4 + y^4 = 16\), there's symmetry about both axes and the origin.

This means if a point \((x, y)\) is on the graph, then the points \((-x, y)\), \((x, -y)\), and \((-x, -y)\) are also on the graph. This symmetry helps in reducing the amount of work needed to plot such curves, as you only need to identify a fraction of the curve and reflect it to find the rest.

Moreover, understanding symmetry in this context helps predict the curve's behavior across all four quadrants, ensuring accuracy and consistency during graphing.

X-Intercepts Calculation

To find where a curve crosses the x-axis, we must calculate its x-intercepts. These occur where the value of \(y\) is zero. For our superellipse, set \(y = 0\) in the equation \(x^4 + y^4 = 16\).

This simplifies to \(x^4 = 16\). Solving this gives us \(x = 2\) and \(x = -2\). Thus, the x-intercepts are the points \((2, 0)\) and \((-2, 0)\).

These calculations show us where the graph touches the x-axis and provide insight into the maximum width of the graph in relation to the axis.

Y-Intercepts Calculation

Finding the y-intercepts means finding where the curve crosses the y-axis by setting \(x = 0\) in the equation. For the superellipse equation \(x^4 + y^4 = 16\), substituting \(x = 0\) gives \(y^4 = 16\).

Solving \(y^4 = 16\) results in \(y = 2\) and \(y = -2\). Therefore, the y-intercepts are the points \((0, 2)\) and \((0, -2)\).

These intercepts define where the graph intersects the y-axis, demonstrating the maximum height and depth of the graph in relation to the y-axis.

Graph Plotting Techniques

Plotting graphs requires careful consideration of intercepts and symmetries. With the equation \(x^4 + y^4 = 16\), begin by marking the intercepts: \((2, 0)\), \((-2, 0)\), \((0, 2)\), and \((0, -2)\).

Utilize symmetry knowledge—given the symmetry across both axes, plot points in one quadrant, and reflect them into the other quadrants, capitalizing on symmetry. This will give you a clear idea of the overall shape.

Finally, double-check by testing additional points to ensure the curves smoothly connect the intercepts. In this case, the overall shape is a smooth, diamond-like curve due to the degree of the terms, making it distinct from circles or ellipses. Practice plotting will build proficiency in accurately representing various shapes of superellipses.