Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ x^{4}+y^{4}=1 $$

Step-by-Step Solution

Answer

The graph is symmetric about the x-axis, y-axis, and origin; intercepts at (1,0), (-1,0), (0,1), and (0,-1); shaped like a rounded square.

Step 1: Check for Symmetries

To determine symmetry about the y-axis, replace \(x\) with \(-x\) in the equation and simplify. The equation becomes \((-x)^4 + y^4 = 1\), which simplifies back to \(x^4 + y^4 = 1\). This indicates even symmetry or symmetry about the y-axis. Similarly, replace \(y\) with \(-y\) to check for symmetry about the x-axis. The equation \(x^4 + (-y)^4 = 1\) simplifies to the original, indicating symmetry about the x-axis. Finally, checking for origin symmetry by replacing both \(x\) with \(-x\) and \(y\) with \(-y\), the equation simplifies to the original, confirming symmetry about the origin.

Step 2: Determine the Intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\). The equation becomes \(x^4 + 0^4 = 1\), which simplifies to \(x^4 = 1\). Taking the fourth root, we get \(x = ±1\). Hence, the x-intercepts are at \((1,0)\) and \((-1,0)\). For y-intercepts, set \(x = 0\), resulting in \(0^4 + y^4 = 1\), or \(y^4 = 1\). Thus, \(y = ±1\), giving y-intercepts at \((0,1)\) and \((0,-1)\).

Step 3: Sketch the Graph

Now plot the x-intercepts and y-intercepts as points on a Cartesian plane: \((1,0), (-1,0), (0,1)\), and \((0,-1)\). Given the symmetry identified in Step 1, draw symmetric curves about the x-axis, y-axis, and the origin. The equation \(x^4 + y^4 = 1\) describes a shape similar to a rounded square with its sides curved inward towards the center, touching these intercept points.

Key Concepts

Graph Symmetry

Graphs can display symmetry in various ways, which helps to simplify and understand their complex structures. For the equation \(x^4 + y^4 = 1\), we examined symmetry by manipulating variables:

• Y-Axis Symmetry: Replace \(x\) with \(-x\). If the equation remains unchanged, it's symmetric about the y-axis. In this case: \((-x)^4 + y^4 = 1\) simplifies back to \(x^4 + y^4 = 1\), confirming y-axis symmetry.
• X-Axis Symmetry: Change \(y\) to \(-y\). Similarly, \(x^4 + (-y)^4 = 1\) reduces to the original, showing x-axis symmetry.
• Origin Symmetry: Substitute both \(x\) and \(y\) with their negatives. The equation \((-x)^4 + (-y)^4 = 1\) again simplifies to \(x^4 + y^4 = 1\), confirming symmetry about the origin.

Recognizing these symmetries allows you to predict and validate the behavior of graphs quickly.

Intercepts

Finding intercepts is an essential skill for graphing equations and understanding their fundamental properties.

• X-Intercepts: Set \(y=0\) and solve for \(x\). In our example, \(x^4 = 1\), which gives \(x = ±1\). The intercepts are \((1,0)\) and \((-1,0)\).
• Y-Intercepts: Set \(x=0\) and solve for \(y\). For this equation, \(y^4 = 1\) gives \(y = ±1\), providing intercepts \((0,1)\) and \((0,-1)\).

These intercepts indicate the points at which the graph meets the axes, guiding us to understand the graph's layout and intersections with the Cartesian plane.

Equation Graphing

Equation graphing translates algebraic expressions into visual representations, enhancing comprehension of behavior and properties. For the equation \(x^4 + y^4 = 1\):

• With the derived intercepts \((1,0), (-1,0), (0,1), (0,-1)\), plot these points on the Cartesian plane.
• Use symmetry properties to anticipate graph shape. Given x- and y-axis as well as origin symmetries, the graph mirrors along these axes.
• Such symmetry suggests a geometric shape like a square but rounded, where all intercepts are part of the graph perimeter.

These guidelines facilitate accurate graph sketching, confirming intercepts and symmetry attributes.

Cartesian Plane

The Cartesian plane is a fundamental tool in calculus for understanding functions and their graphical representations. It consists of two perpendicular number lines that intersect at a point called the origin, divided into four quadrants.

• X-Axis: Horizontal line where y-axis equals zero.
• Y-Axis: Vertical line where x-axis equals zero.
• Each point on the plane corresponds to an ordered pair \((x, y)\).
• The quadrants help in locating points, with the positive x-values to the right, negative x-values to the left, positive y-values above, and negative y-values below.

The Cartesian plane is indispensable for plotting graphs, visualizing symmetries, and determining intercepts through simple graphical analysis.