Unformatted text preview: 1 in Deﬁnition 7.2. Replacing x with (x − h)
and y with (y − k ) causes the usual horizontal and vertical shifts. Replacing x with x and y with y
a
b
causes the usual vertical and horizontal stretches. In other words, it is perfectly ﬁne to think of an
ellipse as the deformation of a circle in which the circle is stretched farther in one direction than
the other.
(x + 1)2 (y − 2)2
Example 7.4.1. Graph
+
= 1. Find the center, the lines which contain the major
9
25
and minor axes, the vertices, and the foci.
Solution. We see that this equation is in the standard form of Equation 7.4. Here x − h is x + 1
so h = −1, and y − k is y − 2 so k = 2. Hence, our ellipse is centered at (−1, 2). We see that a2 = 9
so a = 3, and b2 = 25 so b = 5. This means that we move 3 units left and right from the center
and 5 units up and down from the center to arrive at points on the ellipse. As an aid to sketching,
we draw a rectangle matching this description, called a guide rectangle, and sketch the ellipse
inside this rectangle as seen below on the left. 7.4 Ellipses 423
y y 7
6 4 3 3 2 2 1
−2 5 4 3 6 5 −4 7 1 −1 1 2 ...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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