Write an equation for the ellipse graphed in standard form

Write an equation for the ellipse graphed in standard form:

The Correct Answer and Explanation is:

9 = 1

Explanation:

To write the equation for the ellipse in standard form, we first need to identify its key features from the graph: the center, the orientation (horizontal or vertical), and the lengths of its major and minor axes.

1. Find the Center (h, k):
   The center of the ellipse is the midpoint of its longest diameter, the major axis. By observing the graph, we can see the ellipse’s rightmost point is at (3, -1) and its leftmost point is at (-7, -1). The center is halfway between these two points. We can find the x-coordinate of the center by averaging the x-coordinates of the vertices: h = (3 + (-7)) / 2 = -4 / 2 = -2. The y-coordinate remains -1. Therefore, the center of the ellipse, (h, k), is at (-2, -1).
2. Determine the Orientation and Semi-Major Axis (a):
   Since the ellipse is wider than it is tall, its major axis is horizontal. The standard form for a horizontal ellipse is (x – h)²/a² + (y – k)²/b² = 1. The value ‘a’ represents the length of the semi-major axis, which is the distance from the center to a vertex along the major axis. The distance from the center (-2, -1) to the right vertex (3, -1) is 5 units (3 – (-2) = 5). So, a = 5, and a² = 25.
3. Determine the Semi-Minor Axis (b):
   The value ‘b’ represents the length of the semi-minor axis, which is the distance from the center to a co-vertex along the shorter minor axis. The top point of the ellipse is at (-2, 2) and the bottom point is at (-2, -4). The distance from the center (-2, -1) to the top co-vertex (-2, 2) is 3 units (2 – (-1) = 3). Thus, b = 3, and b² = 9.
4. Write the Equation:
   Now we substitute the values for h, k, a², and b² into the standard equation for a horizontal ellipse:
   (x – h)²/a² + (y – k)²/b² = 1
   (x – (-2))² / 25 + (y – (-1))² / 9 = 1
   (

mos(experimental)

Auto

Expand to view model thoughtschevron_right

21.1seditmore_vert

x + 2)² / 25 + (y + 1)² / 9 = 1

This final equation accurately represents the ellipse shown on the graph.thumb_upthumb_down