general equation of an ellipse
TRANSCRIPT
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