Claudio Alvarado Rylon Guidry Erica Lux Conic sections Claudio Alvarado Rylon Guidry Erica Lux

Look! A square! 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. In order to get a conic into the general equation you must Complete the square to change the equation of y = ax2 +bx +c into the general equation.

Completing the Square Example: y=3x2- 18x –10 Step 1:Isolate the x terms y=3x2- 18x –10 +10 +10 y+10=3x2-18x Step 2: Divide by the x2 coefficient. y+10=3x2-18x 3 3 y+10=x2-6x 3 Step 3: (a) divide the x coefficient by 2 then square it add the product to both sides of the equation y+ 10= x2 +6x 3 -6/2=(-3)2=9 y+10+9= x2-6x+9 3

Completing the square 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)2 -37 -37 y=3(x-3)2 -37 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 This is getting tough!!!

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

Parabola Graph Directrix Focus Parabola

Form of Equation y=a(x-h)2 +k x=a(y-k)2+h Axis of symmetry x=h y=k Vertex (h,k) Focus (h,k+1/4a) (h+1/4a,k) Directrix y=k-1/4a x=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

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= r2 Center of a circle-(h,k) Radius- r Pretty circle!

Circles Find the center and the radius of a circle with and equation of x2+ y2+ 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 x2+ y2+ 2x+ 4y-11=0 x2 + 2x + y2+ 4y =11 Step 2: Complete the Square x2 + 2x + y2+ 4y =11 x2+2x+1+y2+4y+4=11+1+4 Step 3: factor x2+2x+1+y2+4y+4=16 (x+1)2+(y+2)2=16 Center = (-1,-2) Radius= 4

Finding Circle Equations Find the radius using the distance formula D=((x2-x1)2+(y2-y1)2)1/2 D=((5-(-7))2+(-10-1)2)1/2 D=((12)2+(-21)2)1/2 D=(144+441)1/2 D=(585)1/2=24.187 Divide by 2 to find radius=12.093 Write the equation- Center=(-1,.5) r2=146.41 (x+1)2+(y-k)2=146.41 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 (x1+x2, y1+y2)= (h,k) 2 2 (-7+5, 11-10) 2 2

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. 4x2 + 9y2 + 16x -18y -11 = 0

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

Ellipses Find the coordinates of the foci and the length of the major and minor axis. Whose equations is 16x2 + 4y2 = 144 x2 + y2 or x2 + y2 a2 + b2 = 1 b2 + a2 144 144 144 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

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

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

Hyperbola Ahhh! 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.

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

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 a2 - b2 =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

References Glencoe Algebra 2 textbook And I did all this! Glencoe Algebra 2 textbook Internet : www.glencoe.com www.wwfhhh.com Erica’s notes Erica’s house Claudio’s house Where’s Rylon’s name

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!!!!!