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Magazine Chapter 0: Problem 36
Plot the graph of the given equation. $$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$
Short Answer
Expert verified
The graph is an ellipse centered at the origin with vertices at \((0, 3)\), \((0, -3)\), \((2, 0)\), and \((-2, 0)\).
Step by step solution
01
Identify the Equation Type
The equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) is in the standard form of the equation of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). We recognize this because it involves squared terms of \(x\) and \(y\) being equal to 1.
02
Determine the Ellipse Parameters
For the given equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), identify \(a^2 = 4\) and \(b^2 = 9\). Thus, \(a = 2\) and \(b = 3\). Since \(b > a\), the ellipse is vertical, with the longer axis (major axis) being along the y-axis.
03
Locate the Center
The ellipse equation is centered at the origin (0,0) because the equation does not have terms involving \(x-h\) or \(y-k\). Therefore, the center is \((0,0)\).
04
Find the Vertices
Since it is a vertical ellipse, the vertices are given by the points \((0, \pm b)\) and \((\pm a, 0)\). Substituting the values of \(a\) and \(b\), the vertices are \((0, 3)\), \((0, -3)\), \((2, 0)\), and \((-2, 0)\).
05
Plot the Ellipse
Plot the points obtained from the vertices on a coordinate plane and sketch the ellipse shape, keeping in mind that the curve passes through these points, with the ends extending along the x-axis to \(\pm 2\) and along the y-axis to \(\pm 3\). The ellipse is symmetrical about both the x-axis and y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conic Sections
Conic sections are fundamental shapes that emerge from the intersection of a plane with a cone. These shapes are highly significant in mathematics and include ellipses, circles, parabolas, and hyperbolas.
Each type of conic section features distinct properties and equations. These sections form the basis for understanding complex geometric figures and real-life structures.
In terms of equations:
Each type of conic section features distinct properties and equations. These sections form the basis for understanding complex geometric figures and real-life structures.
In terms of equations:
- A circle is a special type of ellipse where the two axes are equal.
- An ellipse has an equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
- A parabola is represented by an equation such as \(y^2 = 4ax\).
- A hyperbola could be written as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Graphing Ellipses
When graphing ellipses, it is essential to recognize their symmetry and shape. The equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) guides us in plotting the curve correctly. Each ellipse consists of two axes: the major axis and the minor axis.
The major axis is the longer one, and it can be either vertical or horizontal, while the minor axis is perpendicular to it.
For graphing an ellipse, you should:
The major axis is the longer one, and it can be either vertical or horizontal, while the minor axis is perpendicular to it.
For graphing an ellipse, you should:
- Identify the center, which in the standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) is at \(h, k\).
- Determine whether it is wider or taller by comparing \(a\) and \(b\).
- Plot the vertices, which define the ends of both axes.
- Sketch the curve ensuring it is smooth and symmetrical about both axes.
Standard Form of an Ellipse
The standard form for the equation of an ellipse is a simple way to describe its shape and features precisely. The general standard form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). In this formula:
In the original equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), \(a^2 = 4\) and \(b^2 = 9\), meaning that the ellipse is centered at \(0,0\) with the center of the ellipse at the origin.
- h and \(k\) are coordinates of the center.
- \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes lengths.
- The ellipse is horizontal if \(a > b\) and vertical if \(b > a\).
In the original equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), \(a^2 = 4\) and \(b^2 = 9\), meaning that the ellipse is centered at \(0,0\) with the center of the ellipse at the origin.
Vertices of an Ellipse
The vertices of an ellipse are crucial points that describe its extremities and help to define its principal axes. For an ellipse centered at \(h, k\), its vertices are given by two sets of coordinates:
In the exercise \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), the vertices are \(0, \pm 3\) and \(\pm 2, 0\), suggesting that the ellipse is taller with its major axis aligned with the y-axis.
Understanding vertices helps in accurately sketching the ellipse and comprehending its dimensions.
- \((h \pm a, k)\) parallel to the x-axis if it is a horizontal ellipse
- \((h, k \pm b)\) parallel to the y-axis if it is a vertical ellipse
In the exercise \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), the vertices are \(0, \pm 3\) and \(\pm 2, 0\), suggesting that the ellipse is taller with its major axis aligned with the y-axis.
Understanding vertices helps in accurately sketching the ellipse and comprehending its dimensions.
