Transcription of General Equation of an Ellipse - University of Minnesota
1 General Equation of an EllipseUniversity of MinnesotaGeneral Equation of an EllipsePreliminaries and ObjectivesPreliminaries Equation of a circle Transformation of graphs (shifting and stretching)Objectives Find the Equation of an Ellipse , given the of MinnesotaGeneral Equation of an EllipseCircle centered at the originxyrxy(x,y)x2+y2=r2x2r2+y2r2=1(xr)2 +(yr)2=1 University of MinnesotaGeneral Equation of an EllipseCircle centered at the originxyrxy(x,y)x2+y2=r2x2r2+y2r2=1(xr)2 +(yr)2=1 University of MinnesotaGeneral Equation of an EllipseCircle centered at the originxyrxy(x,y)x2+y2=r2x2r2+y2r2=1(xr)2 +(yr)2=1 University of MinnesotaGeneral Equation of an EllipseStretching, Period and Wavelengthy=sin(Bx)The sine wave isBtimes thinner. Period (wavelength) is dividedbyB. Frequency is multiplied (xb)The sine wave isbtimes wider. Period (wavelength) ismultiplied byb. Frequency is divided byb.(xr)2+(yr)2=1 The unit circle is stretchedrtimes wider andrtimes of MinnesotaGeneral Equation of an EllipseStretching, Period and Wavelengthy=sin(Bx)The sine wave isBtimes thinner.
2 Period (wavelength) is dividedbyB. Frequency is multiplied (xb)The sine wave isbtimes wider. Period (wavelength) ismultiplied byb. Frequency is divided byb.(xr)2+(yr)2=1 The unit circle is stretchedrtimes wider andrtimes of MinnesotaGeneral Equation of an EllipseStretching, Period and Wavelengthy=sin(Bx)The sine wave isBtimes thinner. Period (wavelength) is dividedbyB. Frequency is multiplied (xb)The sine wave isbtimes wider. Period (wavelength) ismultiplied byb. Frequency is divided byb.(xr)2+(yr)2=1 The unit circle is stretchedrtimes wider andrtimes of MinnesotaGeneral Equation of an EllipseEllipse Centered at the Origin(xr)2+(yr)2=1 The unit circle is stretchedrtimes wider andrtimes taller.(xa)2+(yb)2=1 The unit circle is stretchedatimes wider andbtimes +y2b2=1 University of MinnesotaGeneral Equation of an EllipseEllipse Centered at the Origin(xr)2+(yr)2=1 The unit circle is stretchedrtimes wider andrtimes taller.
3 (xa)2+(yb)2=1 The unit circle is stretchedatimes wider andbtimes +y2b2=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(3,0)( 3,0)(0,5)(0, 5)x29+y225=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(3,0)( 3,0)(0,5)(0, 5)x29+y225=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(a,0)( a,0)(0,b)(0, b)x2a2+y2b2=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(4,0)( 4,0)(0,6)(0, 6)x216+y236=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(4,0)( 4,0)(0,6)(0, 6)x216+y236=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(4,0)( 4,0)(0,6)(0, 6)x216+y236=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(4,0)( 4,0)(0,6)(0, 6)x216+y236=1 University of MinnesotaGeneral Equation of an EllipseEllipse centered at the originxy(4,0)( 4,0)(0,6)(0, 6)x216+y236=1 University of MinnesotaGeneral Equation of an EllipseGeneral Form of an Ellipse (x h)2a2+(y k)2b2=1 Center at(h,k)Vertices at(h+a,k),(h a,k),(h,k+b),(h,k b) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)
4 29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 1 Graph 9(x 3)2+16(y+2)2=144(x 3)216+(y+2)29=1xy( 1, 2)(7, 2)(3,1)(3, 5) University of MinnesotaGeneral Equation of an EllipseExample 2xy( 6, 1)(2, 1)( 2,5)( 2, 7)(x+2)2+(y+1)2=1 University of MinnesotaGeneral Equation of an EllipseExample 2xy( 6, 1)(2, 1)( 2,5)( 2, 7)(x+2)2a2+(y+1)2b2=1 University of MinnesotaGeneral Equation of an EllipseExample 2xy( 6, 1)(2, 1)( 2,5)( 2, 7)(x+2)216+(y+1)2b2=1 University of MinnesotaGeneral Equation of an EllipseExample 2xy( 6, 1)(2, 1)( 2,5)( 2, 7)(x+2)216+(y+1)236=1 University of MinnesotaGeneral Equation of an EllipseRecapGeneral Equation of an Ellipse (x h)2a2+(y k)2b2=1 Center at(h,k)Vertices at(h+a,k),(h a,k),(h,k+b),(h,k b) University of MinnesotaGeneral Equation of an EllipseCreditsWritten by:Mike WeimerskirchNarration:Mike WeimerskirchGraphic Design: Mike WeimerskirchUniversity of MinnesotaGeneral Equation of an EllipseCopyright Infoc The Regents of the University of Minnesota & MikeWeimerskirchFor a license please contact of MinnesotaGeneral Equation of an Ellips
