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Chapter 13: Problem 44

Graph each ellipse. \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)

Short Answer

Expert verified
The ellipse has a semi-major axis of length 4 in the y-direction and a semi-minor axis of length 3 in the x-direction. The vertices are at (0, 4) and (0, -4); the co-vertices are at (3, 0) and (-3, 0).

Step by step solution

01

- Identify the Standard Form

The given ellipse equation is in the standard form of an ellipse centered at the origin:
02

- Identify the Semi-Major and Semi-Minor Axes

In the equation . Here, the Therefore, the lengths of the semi-major axis is in the y -direction The length of the semi-minor axis is in the -direction.
03

- Determine the Vertices and Co-Vertices

Find the vertices located a distance from the center along the axis: Therefore, the vertices are at the points Locate the co-vertices a distance of from the center along the x -axis: Therefore, the co-vertices are at the points e .
04

- Sketch the Ellipse

Plot the vertices , , the co-vertices,, and. Draw a smooth curve through these points forming the ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of an Ellipse
Ellipses have a special equation called the standard form, which makes graphing them much easier. For an ellipse centered at the origin, the standard form is: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes respectively. When the center is not at the origin but at \((h, k)\), the form changes slightly: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Knowing this, it鈥檚 easier to identify and plot the important parts of the ellipse.
Semi-Major Axis
The semi-major axis is one of the most important parts of an ellipse. It is the longest radius of the ellipse. In our example, the equation is \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \). You can see that \(a^2 = 16 \) which means \(a = 4\). The semi-major axis is 4 units long and it lies along the y-axis. So, one end of the semi-major axis is at (0,4) and the other is at (0,-4). In other words, it extends 4 units up and down from the center.
Semi-Minor Axis
The semi-minor axis is the shorter radius of an ellipse. For the equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), you see \(b^2 = 9\), meaning \(b = 3\). The semi-minor axis is 3 units long and lies along the x-axis. This implies that one end of the semi-minor axis is at (3,0), and the other is at (-3,0). It stretches 3 units to the left and to the right from the center.
Vertices
Vertices are the points where the ellipse is widest or tallest. They lie at the ends of the major axis. In our case, the major axis lies along the y-axis. Thus, the vertices are at points where the semi-major axis ends, which are (0,4) and (0,-4). Remember, vertices are always located at a distance of \(a\) units from the center. So, if your ellipse is longer horizontally, your vertices would be along the x-axis instead.
Co-Vertices
Co-vertices are found at the ends of the minor axis. They lie along the axis that is shorter. For the ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), the minor axis is along the x-axis. Hence, the co-vertices are at (3,0) and (-3,0). Co-vertices are always found \(b\) units from the center. Knowing the co-vertices helps in plotting the precise shape and orientation of the ellipse, ensuring a more accurate graph.

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