Example

Example: Graphing a Vertical Ellipse with Center (h,k)(h, k)

To graph the ellipse given by the equation (x+3)24+(y5)216=1\frac{(x+3)^2}{4} + \frac{(y-5)^2}{16} = 1, first identify the center (h,k)(h, k) from the standard form, which is (3,5)(-3, 5). Then, check the denominators to find the orientation of the major axis. Because 16>416 > 4 and 1616 is under the (y5)2(y-5)^2 term, the major axis is vertical. The distance from the center to the vertices along this vertical axis is determined by b2=16b^2 = 16, which gives b=±4b = \pm 4 and a distance of 44. The distance to the endpoints of the minor axis along the horizontal axis comes from a2=4a^2 = 4, resulting in a=±2a = \pm 2 and a distance of 22. To draw the graph, start at the center (3,5)(-3, 5), move 44 units up and down for the vertices, and move 22 units left and right for the minor axis endpoints.

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