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Understanding the standard form of the ellipse equation is essential for graphing and analyzing the properties of an ellipse. The standard form is given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. For a vertically oriented ellipse, the larger value is under the \(y^2\) term, indicating that the major axis is vertical.

When graphing an ellipse, one must first identify \(a\) and \(b\) and use them to find the vertices and co-vertices of the ellipse. For example, in the equation \(25x^2 + 16y^2 = 400\), by dividing both sides by 400 and rearranging, you get \(\frac{x^2}{16} + \frac{y^2}{25} = 1\), from which we can tell that \(a^2 = 25\) and \(b^2 = 16\), giving us \(a = 5\) and \(b = 4\). This step is crucial for understanding the size and orientation of the ellipse.

Ellipse eccentricity, denoted by \(e\), is a measure of how much an ellipse deviates from being a circle. The eccentricity can range from 0 (a perfect circle) to just below 1 (a highly elongated ellipse). To calculate the eccentricity, use the formula \(e = \sqrt{1 - \frac{b^2}{a^2}}\), where \(a\) is the length of the semi-major axis and \(b\) the semi-minor axis.

In our case, with \(a = 5\) and \(b = 4\), the eccentricity is \(e = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}\). This value is less than 1, confirming that the shape is indeed an ellipse. A moderate eccentricity like \(\frac{3}{5}\) indicates that the ellipse is more elongated than a circle, but not extremely so.

The foci of an ellipse are two points located along the major axis, equidistant from the center, that define the shape and extent of the ellipse. The foci are essential to the ellipse's definition; the ellipse is the set of all points for which the sum of the distances to the foci is constant.

For a vertically oriented ellipse, the coordinates of the foci are of the form (0, ±ae). Using our values, we calculate the foci as (0, ±5 * \(\frac{3}{5}\)) which simplifies to (0, ±3). It is important to notice how the eccentricity plays a role in determining the location of the foci. The greater the eccentricity, the further apart the foci are.

The vertices of an ellipse are points where the ellipse intersects the major axis, and they represent the farthest points from the center of the ellipse along that axis. For our vertically oriented ellipse, we expect to see the vertices along the y-axis, given by the format (0, ±a).

Using the previously identified value of \(a = 5\), the vertices of our ellipse are at the points (0, ±5). The vertices are vital for sketching since they outline the maximum bounds of the ellipse along the major axis. When drawing the ellipse, aligning it with the vertices ensures accurate representation of its shape and orientation.