General Equation of an Ellipse
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General Equation of an Ellipse University of Minnesota General Equation of an Ellipse
Transcript of General Equation of an Ellipse
General Equation of an Ellipse
University of Minnesota General Equation of an Ellipse
Preliminaries and Objectives
Preliminaries• Equation of a circle• Transformation of graphs (shifting and stretching)
Objectives• Find the equation of an ellipse, given the graph.
University of Minnesota General Equation of an Ellipse
Circle centered at the origin
x
y
r
x
y
(x , y)
x2 + y2 = r2
x2
r2 +y2
r2 = 1(x
r
)2+(y
r
)2= 1
University of Minnesota General Equation of an Ellipse
Circle centered at the origin
x
y
r
x
y
(x , y)
x2 + y2 = r2
x2
r2 +y2
r2 = 1
(xr
)2+(y
r
)2= 1
University of Minnesota General Equation of an Ellipse
Circle centered at the origin
x
y
r
x
y
(x , y)
x2 + y2 = r2
x2
r2 +y2
r2 = 1(x
r
)2+(y
r
)2= 1
University of Minnesota General Equation of an Ellipse
Stretching, Period and Wavelength
y = sin(Bx)
The sine wave is B times thinner. Period (wavelength) is dividedby B. Frequency is multiplied by B.
y = sin(x
b
)The sine wave is b times wider. Period (wavelength) ismultiplied by b. Frequency is divided by b.
(xr
)2+(y
r
)2= 1
The unit circle is stretched r times wider and r times taller.
University of Minnesota General Equation of an Ellipse
Stretching, Period and Wavelength
y = sin(Bx)
The sine wave is B times thinner. Period (wavelength) is dividedby B. Frequency is multiplied by B.
y = sin(x
b
)The sine wave is b times wider. Period (wavelength) ismultiplied by b. Frequency is divided by b.
(xr
)2+(y
r
)2= 1
The unit circle is stretched r times wider and r times taller.
University of Minnesota General Equation of an Ellipse
Stretching, Period and Wavelength
y = sin(Bx)
The sine wave is B times thinner. Period (wavelength) is dividedby B. Frequency is multiplied by B.
y = sin(x
b
)The sine wave is b times wider. Period (wavelength) ismultiplied by b. Frequency is divided by b.
(xr
)2+(y
r
)2= 1
The unit circle is stretched r times wider and r times taller.
University of Minnesota General Equation of an Ellipse
Ellipse Centered at the Origin
(xr
)2+(y
r
)2= 1
The unit circle is stretched r times wider and r times taller.(xa
)2+(y
b
)2= 1
The unit circle is stretched a times wider and b times taller.
x2
a2 +y2
b2 = 1
University of Minnesota General Equation of an Ellipse
Ellipse Centered at the Origin
(xr
)2+(y
r
)2= 1
The unit circle is stretched r times wider and r times taller.(xa
)2+(y
b
)2= 1
The unit circle is stretched a times wider and b times taller.
x2
a2 +y2
b2 = 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(3,0)(−3,0)
(0,5)
(0,−5)
x2
9+
y2
25= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(3,0)(−3,0)
(0,5)
(0,−5)
x2
9+
y2
25= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(a,0)(−a,0)
(0,b)
(0,−b)
x2
a2 +y2
b2 = 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(4,0)(−4,0)
(0,6)
(0,−6)
x2
16+
y2
36= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(4,0)(−4,0)
(0,6)
(0,−6)
x2
16+
y2
36= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(4,0)(−4,0)
(0,6)
(0,−6)
x2
16+
y2
36= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(4,0)(−4,0)
(0,6)
(0,−6)
x2
16+
y2
36= 1
University of Minnesota General Equation of an Ellipse
Ellipse centered at the origin
x
y
(4,0)(−4,0)
(0,6)
(0,−6)
x2
16+
y2
36= 1
University of Minnesota General Equation of an Ellipse
General Form of an Ellipse
(x − h)2
a2 +(y − k)2
b2 = 1
Center at (h, k)
Vertices at (h + a, k), (h − a, k), (h, k + b), (h, k − b)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 1
Graph 9(x − 3)2 + 16(y + 2)2 = 144
(x − 3)2
16+
(y + 2)2
9= 1
x
y
(−1,−2) (7,−2)
(3,1)
(3,−5)
University of Minnesota General Equation of an Ellipse
Example 2
x
y
(−6,−1) (2,−1)
(−2,5)
(−2,−7)
(x + 2)2+
(y + 1)2= 1
University of Minnesota General Equation of an Ellipse
Example 2
x
y
(−6,−1) (2,−1)
(−2,5)
(−2,−7)
(x + 2)2
a2 +(y + 1)2
b2 = 1
University of Minnesota General Equation of an Ellipse
Example 2
x
y
(−6,−1) (2,−1)
(−2,5)
(−2,−7)
(x + 2)2
16+
(y + 1)2
b2 = 1
University of Minnesota General Equation of an Ellipse
Example 2
x
y
(−6,−1) (2,−1)
(−2,5)
(−2,−7)
(x + 2)2
16+
(y + 1)2
36= 1
University of Minnesota General Equation of an Ellipse
Recap
General Equation of an Ellipse
(x − h)2
a2 +(y − k)2
b2 = 1
Center at (h, k)
Vertices at (h + a, k), (h − a, k), (h, k + b), (h, k − b)
University of Minnesota General Equation of an Ellipse
Credits
Written by: Mike Weimerskirch
Narration: Mike Weimerskirch
Graphic Design: Mike Weimerskirch
University of Minnesota General Equation of an Ellipse
Copyright Info
c© The Regents of the University of Minnesota & MikeWeimerskirchFor a license please contact http://z.umn.edu/otc
University of Minnesota General Equation of an Ellipse
