This activity focuses on understanding the relationship between the equation of an ellipse and its graphical representation. The standard form of an ellipse centered at $(h, k)$ is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. By matching equations to graphs, you'll reinforce your understanding of how these parameters affect the ellipse's shape, orientation, and center. Remember, if $a > b$, the major axis is horizontal; if $b > a$, the major axis is vertical.
The goal is to solidify your ability to quickly identify key features of an ellipse (center, vertices, co-vertices) from its equation and vice versa. This is a fundamental skill for more advanced topics in analytic geometry and calculus.
Match the terms to their definitions:
| Term | Definition |
|---|---|
| 1. Major Axis | A. The point at the center of the ellipse. |
| 2. Minor Axis | B. The longer axis of the ellipse. |
| 3. Center | C. The points on the ellipse farthest from the center. |
| 4. Vertices | D. The shorter axis of the ellipse. |
| 5. Co-vertices | E. The endpoints of the minor axis. |
An ellipse is defined as the set of all points such that the sum of the distances from two fixed points, called _____, is constant. The line through the foci intersects the ellipse at two points called the _____. The other axis is called the _____ axis.
Explain how changing the values of $a$ and $b$ in the standard equation of an ellipse, $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, affects the shape and orientation of the ellipse.
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