Question

Plot the eurve \(\quad x^{4}+y^{4}=81\),

Short answer

Question: Briefly describe the process of graphing the curve for the equation \(x^4+y^4=81\). Answer: To graph the curve, first rewrite the equation in terms of y as \(y=\pm\sqrt[4]{81 - x^4}\). Analyze the behavior of the curve and observe its symmetry with respect to both the x-axis and y-axis. Next, find key points such as (0,3), (0,-3), (3,0), (-3,0) and use symmetry to find corresponding points in other quadrants. Finally, sketch the curve by connecting these key points and ensuring a smooth curve that resembles a squashed square.

Step-by-step solution

Rewrite the equation in terms of y

To rewrite the equation in terms of \(y\), we can isolate \(y^4\) and then take the fourth root of both sides. This will give us: \[y^4 = 81 - x^4\] \[y = \pm\sqrt[4]{81 - x^4}\]

Analyze the behavior of the curve

As \(x\) gets closer to \(0\), \(y\) approaches \(\pm\sqrt[4]{81}\), which is \(\pm3\). Notice that the curve is symmetric with respect to both the \(x\)-axis and the \(y\)-axis. This means that if we find points in the first quadrant, we can easily find corresponding points in the other three quadrants.

Find key points

Here are some key points to plot: 1. \((x,y) = (0,3)\) and \((x,y) = (0,-3)\): These points occur when \(x = 0\). 2. \((x,y) = (3,0)\) and \((x,y) = (-3,0)\): These points occur when \(y = 0\). 3. \((x,y) = (1,\sqrt{80})\) and \((x,y) = (1,-\sqrt{80})\): These points occur when \(x = 1\). Using symmetry, we could find points in the other quadrants: e.g., for \((x,y)=(1,\sqrt{80})\), the corresponding points in the other three quadrants are \((-1,\sqrt{80})\), \((-1,-\sqrt{80})\), and \((1,-\sqrt{80})\). Plot these points as well as other key points to create a sketch of the curve.

Sketch the curve using key points and symmetry

With a clear understanding of the behavior of the curve for various values of \(x\), we can now sketch the curve. Start by plotting the key points we identified above and use the symmetry properties to find corresponding points in other quadrants. Then, connect the points to create a smooth curve. The curve will resemble a squashed square that touches the x-axis at \((\pm 3,0)\) and y-axis at \((0,\pm 3)\). Keep the shape smooth as it curves around these points. After plotting the curve and verifying its properties, you should have a good understanding of the graph of the given equation \(x^4+y^4=81\).

Key concepts

Graphing Equations

Graphing equations is a crucial part of understanding mathematical relationships visually. In this exercise, we look at the equation \(x^4 + y^4 = 81\). To plot this graph, we need to identify various key components, such as key points and their behavior. By analyzing the equation and identifying values for \(x\) and \(y\), we can generate points on a graph. Breaking down the original equation allows us to predict its shape and understand the curve better.

The function \(y = \pm\sqrt[4]{81 - x^4}\) describes the relationship between \(x\) and \(y\) by isolating \(y\). This equation gives us a sense of the curve's structure, which helps us find points that belong to the graph. Once the equation is simplified, we can substitute different \(x\)-values into the equation to find corresponding \(y\)-values that form coordinates. It's useful to choose simple values to start with, as they reveal the nature and symmetry of the graph. Plotting these points leads to a clear sketch of the curve.

One of the most important aspects is understanding where the curve meets the axes. For example, when \(x=0\), \(y\) can be \(\pm 3\), indicating intersections on the \(y\)-axis. Similarly, when \(y=0\), \(x\) can be \(\pm 3\), marking x-axis intersections. Graphs reveal more than numerical answers because they offer a visual insight into how variables interact in an equation.

Symmetry in Graphs

Symmetry is a powerful tool in graphing and can simplify our understanding of complex equations. For the equation \(x^4 + y^4 = 81\), symmetry drastically reduces the effort needed to plot the curve.

There are two main symmetries in this curve; one is symmetry with respect to the y-axis and the other with respect to the x-axis. What this means is if there is a point \((x, y)\), there will also be points \((-x, y)\), \((x, -y)\), and \((-x, -y)\). This symmetry leads to the curve being identical in all quadrants, reflecting across both axes.

• The symmetry about the y-axis implies that the graph is a mirror image for positive and negative \(x\) values.
• The symmetry about the x-axis shows the same mirror image property for positive and negative \(y\) values.

By understanding these properties, we only need to plot points in one quadrant to know the entire graph.

Finding symmetry helps us make connections between plotted points without having to evaluate the entire equation repeatedly. This cuts down on calculation and ensures that our graph is plotted accurately in each quadrant, enhancing our comprehension of overall graph behavior.

Quadrant Analysis

Quadrant analysis allows us to determine where specific points will lie on a graph, based on the signs of \(x\) and \(y\) coordinates.

In our exercise with \(x^4 + y^4 = 81\), analyzing each quadrant reveals more about the graph's coverage across the coordinate plane. The Cartesian Plane divides the graph into four quadrants:

• Quadrant I: Both \(x\) and \(y\) are positive.
• Quadrant II: \(x\) is negative and \(y\) is positive.
• Quadrant III: Both \(x\) and \(y\) are negative.
• Quadrant IV: \(x\) is positive and \(y\) is negative.

⠀Since the graph is symmetric with respect to both axes, finding solutions for one quadrant helps identify solutions in the other three quadrants. With this understanding, points such as \((1, \sqrt{80})\) in Quadrant I immediately give us additional points like \((-1, \sqrt{80})\), \((-1, -\sqrt{80})\), and \((1, -\sqrt{80})\) in the other quadrants. This practical approach to quadrant analysis helps us 'mirror' points correctly across the graph, avoiding errors and saving time during graphing tasks.

By mastering quadrant analysis, you'll be able to draft complete and accurate graphical representations of different polynomial curves, unlocking a deeper understanding of multi-variable equations.