91Ó°ÊÓ

When graphing an ellipse, it's essential to recognize the standard form equation of an ellipse. This equation is crucial to identifying the key features of the ellipse, such as its center, axes, and overall shape. The standard form for the equation of an ellipse with a horizontal major axis is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) and the form for a vertical major axis is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\), where \(h, k\) represent the center of the ellipse.
The denominators \(a^2\) and \(b^2\) provide us with the lengths of the major and minor axes, specifically \(2a\) and \(2b\) respectively. If \(a^2 > b^2\), the major axis is horizontal; if \(b^2 > a^2\), it's vertical. Understanding this form will greatly assist in accurately graphing the ellipse, pinpointing every aspect related to its geometry.
In the given exercise, the equation \(\frac{(x+3)^{2}}{9}+(y-2)^{2}=1\) is already in standard form, making it easier to proceed with the graphing process. Recognizing that the larger denominator is with the \(x\)-term, we can determine that we're dealing with a horizontal ellipse.

The foci (plural of focus) of an ellipse are two fixed points located along the major axis that are equidistant from the center. The unique property of the foci is that the sum of the distances from the foci to a point on the ellipse is constant, and is equal to the length of the major axis.
To find the foci, you can use the formula \(c = \sqrt{a^2 - b^2}\), where \(c\) is the distance from the center to each focus, \(a\) is half the length of the major axis, and \(b\) is half the length of the minor axis. This formula arises from the constant sum property and the Pythagorean theorem applied to the right triangle formed by the foci and any point on the ellipse.
In our exercise, using \(a=3\) and \(b=1\), we find that \(c = \sqrt{9 - 1} = \sqrt{8}\), which approximates to about 2.83. This reveals the exact position of the foci on a horizontal line passing through the center at \(y=2\), horizontally \(\sqrt{8}\) units away from the center at \(x=-3\).

The major axis of an ellipse is the longest diameter that runs through the center and across the two furthest points on the curve. The minor axis is the shortest diameter, also through the center, but perpendicular to the major axis. Ideally, these axes cross one another at the center point of the ellipse and define its width and height.
The lengths of these axes can be directly deduced from the standard form equation of the ellipse. They are vital to plotting the correct shape, as they determine the 'stretch' of the ellipse along the horizontal and vertical directions.
Let's apply this to the given problem. Since we identified the major axis as horizontal with a length of \(6\) units (\(2\sqrt{9}\)) and a minor axis length of \(2\) units (\(2\sqrt{1}\)), we recognize the ellipse's proportions. Drawing the axes through the center \( (-3, 2) \) allows us to sketch the overall figure ensuring that all points are equidistant from the foci via the ellipse's constant sum property.