Quadratic Relations and Conic Sections Chapter 8 Review Quadratic Relations and Conic Sections Gracie Molnar, Maggie Cook, Rose Harris-Forkner, Mackenzie Mencias, Kelly Skwarcan, and Mary Rice

Solving Quadratic Systems: Substitution Solve one equation for y. Substitute this equation in for y in the second equation. Solve for x. Substitute this x value into the original equation and solve for y. Write the x and y values as an ordered pair.

Solving Systems: Substitution Example x2 + y2- 9 = 0 y - 4x = 12 Solve equation for y. y = 4x + 12 Substitute into other equation: x2 + (4x+12)(4x+12) - 9 = 0 Simplify. 17x2 + 96x + 135 = 0 Factor. (17x+45)(x+3) = 0 Solve for x. x= -45/17, - 3 Substitute for x in the original equation. y=4(-45/17) +12 y=4(-3) +12 Solve for y. y=24/17 y=0 Write solutions as data points. (-45/17, 24/17) & (-3, 0)

Solving Systems: Elimination Steps: Find the LCF of the coefficients of either the x or y variable. Multiply one or both of the equations by the LCF in order to cancel out one variable. Solve for the other variable. Substitute this variable’s value into one equation. Solve for the variable. Write the solutions as a data point.

Solving Systems: Elimination Example 3y + 8x = 2 y + 2x = 12 Multiply one or more equations by a factor that can eliminate a variable. Add the equations together and solve. 2x = -34 → x= -17 Substitute the x value into an equation. y + 2(-17) = 12 Solve for y. y = 46 Write solutions as an ordered pair. (17, 46) 3y + 8x = 2 → 3y + 8x = 2 -3(y + 2x = 12) → -3y - 6x = -36

What is the formula of a circle? (x-h)2 + (y-k)2 = r2

Circles Example: Circle Equation: (x-h)2+(y-k)2=r2 (h,k)= coordinates of center *When you’re trying to decide what conic section you have based on an equation, note that a circle has both variables squared, the coefficients are the same, and you are able to factor the equation to get the standard form. Radius=2 Center= (0,0) (x2)+(y2)=4 radius

Distance and Midpoint Formulas Distance formula: (x2-x1)2+(y2-y1)2 Midpoint formula: For the x coordinate: x1+x2 2 For the y coordinate: y1+y2

What is the formula of a vertical parabola? (x-h)2 = 4p(y-k)

Parabola Formulas Vertical Standard form: (x-h)2=4p(y-k) Vertex: (h,k) AOS: x=h Focus: (h, k + p) Directrix: y=k - p Horizontal Standard Form: (y - k)2=4p(x-h) Vertex: (h,k) AOS: y=k Focus: (h + p, k) Directrix: x=h - p

Focus: (h+p,k) (-⅛,0) Directrix: x=h-p x=1/8 Parabolas Graph the following parabola and include all necessary characteristics. x=-2y2 Vertex: (0,0) AOS: y=k y=0 Focus: (h+p,k) (-⅛,0) Directrix: x=h-p x=1/8 Finding p: Isolate squared variable and set non-squared variable =4p p= -⅛

Find the directrix of the following equation: Parabolas Continued Find the directrix of the following equation: -y2=18x X=h-p y^2=18x → -18=4p → p=9/2 x=0-(-9/2) X=9/2

What is the formula of a horizontal ellipse? (x-h)2 + (y-k)2 =1 c2 b2

Ellipses Horizontal Characteristics Horizontal: (x-h)2+(y-k)2 =1 c2 b2 Center: (h,k) Vertices: (h±c,k) Foci: (h±a,k) Co-vertices: (h,k±b) Ellipses Horizontal: (x-h)2+(y-k)2 =1 c2 b2 Vertical: (x-h)2 + (y-k)2 = 1 b2 c2 Vertical Characteristics Center: (h,k) Vertices: (h,k±c) Foci: (h,k±a) Co-vertices: (h±b,k)

How to graph a horizontal ellipse x2 + y2 = 1 100 64 Note: Since the first denominator is larger, the ellipse is horizontal. ___ √100 = c ±10 = c __ √64 = b ±8 = b Center:(0 , 0) Vertices:(10 , 0) & (-10 , 0) Foci:(6 , 0) & (-6 , 0) Co-vertices:(0 , 8) & (0 , -8) a2+b2=c2 a2+64=100 ______ a=√36 a=6

What is the formula of a vertical hyperbola? (y-k)2 - (x-h)2 =1 a2 b2

Hyperbolas Characteristics: Horizontal Vertical Horizontal: (h,k) Center (h,k) (h+a,k) Vertices (h, k+ a) (h+c,k) Foci (h, k+c) + b Slope of Transverse + a a Axis b Horizontal: (x-h)2 - (y-k)2 = 1 a2 b2 Vertical: (y-k)2 - (x-h)2 =1 a2 b2

Hyperbola Example *Remember to find other x or y values: if it’s a horizontal hyperbola, move one x value left or right from the vertices. If it’s a vertical hyperbola, move one y value up or down from the vertices. Then, plug this value back into the equation and solve for the other variable to find the other values. Graph the following hyperbola: (x-1)2 - (y+4)2 = 1 9 16 First, determine if it is horizontal or vertical -Since this hyperbola has x first, it is horizontal. Then determine the characteristics -a=3 b=4 c=5 -Center: (h,k)-->(1, -4) -Vertices:(h+a,k)--> (4, -4) (-2, 4) -Foci:(h+c,k)-->(6, -4) (-4, -4) -Slope of Transverse Axis: + b + 4 a 3

Hyperbola Graph Plot the points in the following order: Transverse Axis Plot the points in the following order: Center Vertices Foci Slope of transverse axis Other points x y 5 -0.47 5 -7.53 -3 -0.47 -3 -7.53 Focus Vertex Center

Classifying Conic Sections If the coefficients in front of the x2 and y2 are the same value and the variables are being added together, then the equation is a circle. If the coefficients are different, but x2 and y2 are being added together, then the equation is an ellipse. If the coefficients are different, but x2 and y2 are being subtracted from each other, then the equation is a hyperbola. If only one variable is squared, then the equation is a parabola.

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