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Conic sections Claudio Alvarado Rylon Guidry Erica Lux

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Complete the Square Parabolas as well as other conic sections are not always in the general form. The general equation is Y=a(x-h) 2 + k.Parabolas as well as other conic sections are not always in the general form. The general equation is Y=a(x-h) 2 + k. In order to get a conic into the general equation you must Complete the square to change the equation of y = ax 2 +bx +c into the general equation.In order to get a conic into the general equation you must Complete the square to change the equation of y = ax 2 +bx +c into the general equation. Look! A square!

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Completing the Square Example: y=3x x –10 Step 1:Isolate the x terms y=3x x – y+10=3x 2 -18x Step 2: Divide by the x 2 coefficient. y+10=3x 2 -18x 3 3 y+10=x 2 -6x 3 Step 3: (a) divide the x coefficient by 2 then square it add the product to both sides of the equation y+ 10= x 2 +6x 3 -6/2=(-3)2=9 y+10+9= x 2 -6x+9 3

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Completing the square Step 4: Factor the right hand side of the equation. y+10+9= x2-6x+9 3 y+10+9=(x-3)(x-3) 3 y+10+9=(x-3)2 3 Step 5: Solve for y do that y=a(x-h) 2 +k y+10+9=(x-3) 2 3 3{y+10}=3(x-3) 2 3 y+10+27=3(x-3) 2 y+37= 3(x-3) y=3(x-3) This is getting tough!!!

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Parabolas Parabola-a set of all points in a plane that are the same distances from a given point called the focus and a given line called the directrix Latus Rectum- the line segment through focus and perpendicular to the axis of symmetry

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Parabola Graph Directrix Focus Parabola

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Form of Equationy=a(x-h)2 +kx=a(y-k)2+h Axis of symmetryx=hy=k Vertex(h,k) Focus(h,k+1/4a)(h+1/4a,k) Directrixy=k-1/4ax=h-1/4a Direction of opening Upward if a>0 Down if a<0 Right if a>0 Left if a<0 Length of Latus Rectum Abs (1/a) units

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Circles Circle- the set of all points in a plane that are equal distances from a given point in the plane called the center. Radius-any segments whose endpoints are the center and a point on the circle Equation of a circle: (x-h) 2 + (y-k) 2 = r 2 Center of a circle-(h,k) Radius- r Pretty circle!

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Circles Find the center and the radius of a circle with and equation of x 2 + y 2 + 2x+ 4y-11=0 Step 1: Put all like terms together on the left hand side of the equation; place on constants on the right x 2 + y 2 + 2x+ 4y-11=0 x 2 + 2x + y 2 + 4y =11 Step 2: Complete the Square x 2 + 2x + y 2 + 4y =11 x 2 +2x+1+y 2 +4y+4= Step 3: factor x 2 +2x+1+y 2 +4y+4=16 (x+1) 2 +(y+2) 2 =16 Center = (-1,-2) Radius= 4

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Finding Circle Equations Write an equation of a circle whose endpoints of its diameter are at (-7,11) and (5,-10) Step 1: Find the center by recalling the midpoint formula (x 1 +x 2, y 1 +y 2 )= (h,k) 2 2 (-7+5, 11-10) 2 Find the radius using the distance formula D=((x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 ) 1/2 D=((5-(-7)) 2 +(-10-1) 2 ) 1/2 D=((12) 2 +(-21) 2 ) 1/2 D=( ) 1/2 D=(585) 1/2 = Divide by 2 to find radius= Write the equation- Center=(-1,.5) r 2 = (x+1) 2 +(y-k) 2 =146.41

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Definition of an Ellipse An ellipse is the set of all points in a plane such that the sum of the distances form the foci is constant. 4x 2 + 9y x -18y -11 = 0

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Ellipses Standard Equation for a center (0,0) A)x2 + y2 a2 + b2 =1 Major Axis isx because a under x Foci (c,o) (-c,o) a2 >b2 b2 = a2 –c2 B)x2 + y2 b2 + a2 =1 Major Axis is y because a under y foci (o,c) (o,-c) True for both equations Take me to your Ellipses

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Ellipses Find the coordinates of the foci and the length of the major and minor axis. Whose equations is 16x2 + 4y2 = 144 x 2 + y 2 or x 2 + y 2 a 2 + b2 = 1 b 2 + a 2 16x 2 + 4y 2 = Since we know a2>b2 major axis is y c=(27) 1\2 c=(9) 1\2 c=3(3) 1\2 Length of your major axis= 2a =12 Length of your minor axis =2b =6 Foci (0,3(3)^1\2) (0,-3(3)^1\2) b2 = a2 – c2 -27 = -c2 9 = 36 – c2 c2 = 27

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Ellipses When the center is Not at the origin (0,0) center(h,k) Standard equation A) (x-h) 2 Ahh!!! Big Big ellipse!!

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Hyperbola Definition A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points, called the foci, is constant

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Hyperbola Standard Equations of Hyperbolas with Center at the Origin If a hyperbola has foci at (-c,o) and (c,o0 and if the absolute value of the difference of the distances from any point on the hyperbola to the tow foci is 2a units, then the standard equation of the hyperbola is x2 - y2 a2 - b2 =1, where c2 = a2+b2. If a hyperbola has foci at (o,-c) and (o,c) and if the absolute value of the difference of the distances from any point on the hyperbola to the two foci is 2a unit, and then the standard equation of the hyperbola is y2 - x2 a2 – b2 = 1, where c2= a2 + b2. Ahhh!

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Hyperbola Equation of Hyperbola x2 – y2 a2 b2=1 y2 – x2 a2 b2 =1 Equation of Asympote b Y=+/- ax a y = +/- bx Transverse Axis horizontalvertical

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Hyperbola Standard Equations of Hyperbolas with Center at (h,k) The equation of a hyperbola with center at (h,k) and with a horizontal transverse axis x-h) 2 - (y-k) 2 a 2 - b 2 =1 The equation of a hyperbola with center at (h,k) and with a vertical transverse axis is (y-k)2 - (x-h)2 a2 - b2 =1

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References References Glencoe Algebra 2 textbook Internet : Ericas notes Ericas house Claudios house WheresRylons name name And I did all this!

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Roles Erica – Poster manager keeper dudette Rylon – real life picture getter dude Claudio – with the help of Erica, did this wonderful presentation for you to behold This marvelous project deserves a 100!!!!!

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