Problem 15
Question
Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}+4 z^{2}-36 y=0 $$
Step-by-Step Solution
Verified Answer
The equation represents a parabolic cylinder opening along the \(y\)-axis.
1Step 1: Rearrange the Equation
To understand the nature of the given equation, start by rearranging it in a more recognizable form. Begin by rewriting the original equation: \( 9x^2 + 4z^2 - 36y = 0 \). Add \(36y\) to both sides to obtain: \( 9x^2 + 4z^2 = 36y \).
2Step 2: Divide to Simplify
Next, divide every term in the equation by 36 to simplify: \( \frac{9x^2}{36} + \frac{4z^2}{36} = \frac{36y}{36} \). This simplifies to \( \frac{x^2}{4} + \frac{z^2}{9} = y \).
3Step 3: Analyze the Equation
The equation \( \frac{x^2}{4} + \frac{z^2}{9} = y \) is now in the standard form of a parabolic cylinder. The presence of quadratic terms in \(x\) and \(z\) and the linear term in \(y\) indicates it is a parabolic equation that opens along the \(y\)-axis.
4Step 4: Identify the Shape and Its Orientation
The graph of the equation represents a parabolic cylinder. In three-space, a parabolic cylinder is defined by a parabola in one cross-section that is extended parallel to an axis—in this case, the \(y\)-axis. The traces in planes parallel to the \(xy\) and \(yz\) planes will be parabolas, confirming the shape as a parabolic cylinder opening along the \(y\)-axis.
5Step 5: Sketch the Graph
To sketch the graph, draw a parabola in the \(xz\)-plane (when \(y=0\)) that resembles \(\frac{x^2}{4} + \frac{z^2}{9} = 0\). This is a degenerate graph (meeting only at origin), which indicates that the main graph develops along the \(y\)-axis as \(y\) increases. Extend this parabola infinitely in the positive and negative \(y\) directions. Thus, the graph is a parabolic cylinder extending along the \(y\)-axis.
Key Concepts
Three-Space GraphsQuadratic EquationsParabolic EquationsStandard Form Transformation
Three-Space Graphs
In mathematics, three-space graphs provide a way to visualize geometric shapes in three-dimensional space. This is unlike the simple x-y graphs you might be familiar with from two-dimensional algebra and geometry. Understanding three-space allows us to represent relationships and equations in a way that includes depth, not just width and height.
This additional dimension can help to visualize complex relationships, like the one described by the given equation \(9x^2 + 4z^2 - 36y = 0\). The three axes, typically labeled x, y, and z, form the foundation of three-space and provide a way to plot graphs that add depth to visualization.
Within three-space, various shapes like spheres, cones, and cylinders—just to name a few—can be represented. This kind of graph is particularly helpful when dealing with equations that involve more than one variable, as it allows you to see how changes in one variable might influence another. In our specific case, the problem involves a parabolic cylinder, a unique shape that extends infinitely in three-dimensional space.
This additional dimension can help to visualize complex relationships, like the one described by the given equation \(9x^2 + 4z^2 - 36y = 0\). The three axes, typically labeled x, y, and z, form the foundation of three-space and provide a way to plot graphs that add depth to visualization.
Within three-space, various shapes like spheres, cones, and cylinders—just to name a few—can be represented. This kind of graph is particularly helpful when dealing with equations that involve more than one variable, as it allows you to see how changes in one variable might influence another. In our specific case, the problem involves a parabolic cylinder, a unique shape that extends infinitely in three-dimensional space.
Quadratic Equations
Quadratic equations play a vital role in understanding a range of mathematical principles. At its core, a quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is an unknown variable.
Quadratics show up in many areas of mathematics, including algebra, calculus, and geometry, serving as a bridge between these disciplines. They often describe parabolic shapes, such as the path of a projectile or, in our case, the sections of a parabolic cylinder.
In the equation \(9x^2 + 4z^2 - 36y = 0\), you can see quadratic elements in the terms \(9x^2\) and \(4z^2\), which are part of transforming the equation into its standard form. Understanding this helps when it comes to graphing these equations in three-space.
Quadratics show up in many areas of mathematics, including algebra, calculus, and geometry, serving as a bridge between these disciplines. They often describe parabolic shapes, such as the path of a projectile or, in our case, the sections of a parabolic cylinder.
In the equation \(9x^2 + 4z^2 - 36y = 0\), you can see quadratic elements in the terms \(9x^2\) and \(4z^2\), which are part of transforming the equation into its standard form. Understanding this helps when it comes to graphing these equations in three-space.
Parabolic Equations
Parabolic equations are a specific type of quadratic equation. They are vital for graphing shapes like parabolas or cylinders, which are structures you'll often see in real-world applications. To grasp parabolic equations, it's helpful to understand that they represent a curve that opens in one direction.
In our current context, the equation more simply written as \(\frac{x^2}{4} + \frac{z^2}{9} = y\) is a parabolic equation in three-dimensional space. This equation describes a parabolic cylinder that opens along the y-axis because the quadratic terms \(x^2\) and \(z^2\) are set equal to the linear y-term.
In our current context, the equation more simply written as \(\frac{x^2}{4} + \frac{z^2}{9} = y\) is a parabolic equation in three-dimensional space. This equation describes a parabolic cylinder that opens along the y-axis because the quadratic terms \(x^2\) and \(z^2\) are set equal to the linear y-term.
- In the xy-plane and yz-plane, traces of the graph will show parabolic shapes.
- The primary axis of extension for this shape is along the y-axis, indicated by the equation's structure.
Standard Form Transformation
Transforming equations into their standard form is crucial for recognizing the kind of graph you are dealing with. Standard form provides a clearer view of the equation's properties and enables straightforward graphing. This is particularly true in cases involving parabolas or other quadratic forms.
Starting with the original equation, \(9x^2 + 4z^2 - 36y = 0\), it was rearranged by adding \(36y\) to both sides to isolate the quadratic components. By dividing everything by 36, the equation becomes \(\frac{x^2}{4} + \frac{z^2}{9} = y\). This transformation simplifies the equation into a recognizable form that denotes a parabolic cylinder.
Starting with the original equation, \(9x^2 + 4z^2 - 36y = 0\), it was rearranged by adding \(36y\) to both sides to isolate the quadratic components. By dividing everything by 36, the equation becomes \(\frac{x^2}{4} + \frac{z^2}{9} = y\). This transformation simplifies the equation into a recognizable form that denotes a parabolic cylinder.
- It helps to identify the axis along which the graph extends, which is the y-axis in our case.
- Enables easier drawing and comprehension of the graph's structure and behavior.
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