Unit 9: Conic Sections Name Per. Test Part 1


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1 Unit 9: Conic Sections Name Per 1/6 HOLIDAY 1/7 General Vocab Intro to Conics Circles 1/89 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics in standard form Parameter changes *Real World Applications HW: Part 1/1516 Review Test Part 1 HW: Part 3 1/17 Test Part HW: Part 5 Objectives: Describe the conic section formed by the intersection of a double right cone and a plane. Determine the vertex form of a quadratic given the standard form Recognize how parameter changes affect the sketch of a conic section. Identify symmetries of conic sections Identify the conic section from an equation Graph and write the equation of the conic section Use the method of completing the square to change the form of the equation Determine the effects of changing a, h, or k on graphs of quadratics (horizontal and vertical) *Graph and write equations of conic sections in real world applications Essential Questions: Compare and contrast the conic equations. How do you know if the conic is vertically or horizontally orientated? General Vocabulary: Know the vocabulary listed below in context to the material learned throughout this unit Double ended right cone Plane Conic (conic section) Intersection Base of a cone Side of a cone Parallel Perpendicular Vertical Horizontal Orientation Symmetry
2 Part 1  Intro to Conics Describe the conic section and its symmetries formed by the intersection of a double right cone and a plane. Name Conic Equation Symmetry: (AOS) Plane Intersection: Real World Part 1  Circles: ( h) +( ) = Graph a circle given the equation: 1. Determine the center from the conic form of the equation (watch for signs +/ h is opposite) (h, k). Find the radius r 3. Graph center and 4 points 1. Graph: (x+5) + (y4) = 5. Write the equation: Center at (0,4) radius of 3 Write the equation of a circle given the graph/info: 1. Find the center (watch signs +/) (h, k). Find the radius r 3. Plug (h, k) & r into conic form of circle equation and simplify Write the equation of a circle in Conic form (C.T.S.): 1. Look for squared variable and group in sets. Move everything else to the other side of the = 3. C.T.S. for any sets of variables of the left 3. Write the equation: 4. x + y x  4y 4 = 0 5. x + y 6x + 4y + 9 = 0
3 Part  Ellipses: ( ) + ( ) =1 Minor Axis shorter diameter of the ellipse Major Axis longer diameter of the ellipse Horizontal Orientation horizontal major axis/ is larger/horz. symm. Vertical Orientation vertical major axis/ is larger/vert. symm. Vertices the points where the major axis touches the ellipse Covertices the points where the minor axis touches the ellipse Focus (foci) fixed points on an ellipse ( ±c, 0) using = Graph an ellipse given the equation: 1. Determine the center from the conic form of the equation (watch signs +/ h is opposite) (h, k). Find the square root of and 3. Count a spaces from center left/right  make point 4. Count b spaces from center up/down make point 5. Connect those 4 points in an elliptical shape 1. Find all parts to the ellipse and graph: ( x 5) ( y + ) 16 9 Center: a =, b =, c = Foci: Vertices: CoVertices: Write the equation of an ellipse given the graph/info: 1. Find the center (h, k) plug it into the equation. Find the a value (always 1st/under x value) count spaces from center left /right in equation 3. Find the b value (always nd/under y value) count spaces from center up/down in equation 4. Always + in between terms and always = 1. Find all parts to the ellipse and graph: ( x + 3) y 4 5 Center: a =, b =, c = Foci: Vertices: CoVertices: 3. Write the equation from graph: *Pre AP Write the equation of an ellipse in Conic form: 1. Group sets of variables on left of =. C.T.S. for both sets variables 3. Pull out GCF /apply to blanks on right side *Example: 9x + 4y  54x  8y 59 = 0
4 Part 3  Hyperbolas: ( ) ( ) =1 Horizontal Orientation hyperbola opens left and right/ is larger/horizontal symmetry Vertical Orientation hyperbola opens up and down/ is larger/ vertical symmetry Vertices the points hyperbola is drawn through Focus (foci) fixed points inside the hyperbola curves ( ±c, 0) using = Asymptotes lines where the graph does not exist the graph cannot cross these lines at any time (used to help guide the drawing of the graph) Equations of Asymptotes Horizontal: = Vertical: = Graph a hyperbola given the equation: 1. Determine the center from the conic form of the equation (watch signs +/ h is opposite) (h, k). Find the square root of and 3. Count a spaces from center left/right  make point 4. Count b spaces from center up/down make point 5. Make a dotted box using the a and b spacing and connect corners to draw in asymptotes 6. Draw in parabola graphs through vertex and approaching asymptotes 1. Find all parts to the hyperbola and graph:. Find all parts to the hyperbola and graph: ( x 3) ( y + ) = y ( x 1) = Center: a = b = Center: a = b = *Pre AP ONLY* Write the equation of a hyperbola given the graph/info: 1. Find (h, k) where the asymptotes intersect equidistant from vertices plug it into the equation. Find the a value (always 1st/under x value) count spaces from center left /right in equation 3. Find the b value (always nd/under y value) use asymptote equations to solve for b (ie: top or bottom of slope at given a value) in equation 4. Always in between terms and always = 1 3. Write the equation from graph: 4. Write the equation from graph:
5 Part 4  Parabolas: Center Vertex of the parabola (h, k) p the distance between the vertex and the focus/directrix determines how the parabola opens Directrix line perpendicular to the axis of symmetry for the parabola p units from vertex Focal Width  length of vertical or horizontal line that passes through the focus and touches parabola on each end Focus  a point on inside the parabola p units from the vertex used to define curve Horizontal Orientation y term is squared opens left/right Vertical Orientation x term is squared opens up/down EOLR points p units up/down from the focus and on the parabola *Note: x and h always stay together & y and k stay together Graph a vertical parabola given the equation: ( h) =4 ( ) 1. Determine if the parabola is vertical or horizontal: a. x term is squared : opens up/down b. p is + opens up : p is opens down. Determine the center from the conic form of the equation (watch signs +/ h is opposite) (h, k) 3. Find p (divide constant by 4) 4. Count p spaces from center up/down  make point (focus) draw in directrix 5. Count p spaces right/left from focus draw ELOR s 6. Draw parabola through vertex and ELOR s Vertical Horizontal Graph a horizontal parabola given the equation: ( ) =4 ( h) 1. Determine if the parabola is vertical or horizontal: a. y term is squared : opens right/left b. p is + opens right : p is opens left. Determine the center from the conic form of the equation (watch signs +/ h is opposite) (h, k) 3. Find p (divide constant by 4) 4. Count p spaces from center left/right  make point (focus) draw in directrix 5. Count p spaces up/down from focus draw ELOR s 6. Draw parabola through vertex and ELOR s 1. Find all parts to the parabola and graph: ( y + ) = 16 Vertex: P = Focus: Directrix: Focal width: EOLR: x. Find all parts to the parabola and graph: ( x ) = 0( y + 1) Vertex: P = Focus: Directrix: Focal width: EOLR: 3. Write in conic form: y y x = 0 4. *PreAP* Write the Equation given limited information a. Focus: (0, 3) Directrix: y = 3 b. Focus: (0, 4) Directrix: x = 3
6 Part 5 Determine the conic from standard form Use to determine the conic: Circle: 4 0 and b = 0 and a = c Ellipse: 4 0 and b 0 OR a c Hyperbola: 4 0 Parabola: 4 = 0 Standard form: = 1. Determine the conic and explain your reasoning: 6x + 9y + 1x 15y 5 = 0 Easy hints: Circle: A = C Ellipse: A C, A & C have same signs Hyperbola: A C, A & C have opposite signs Parabola: A = 0 or C = 0, NOT both (just an x or just a y ). Determine the conic and explain your reasoning: x + y 6x 7 = 0 3. Determine the conic and explain your reasoning: 3y x 9 = 0 4. Determine the conic and explain your reasoning: y x 4y + 10 = 0 Part 5 Parameter changes with conics For ALL h shifts left/right k shifts up/down Circle r changes radius Ellipse a & b change size and direction a b b a Parabola p determines whether the parabola opens up/down/left/right p determines wider or narrower 1. If the center of an ellipse if shifted to the right by 4 which value is changed?. If the size a circle increases, what value is changed? 3. If a and b are switched in the equation of an ellipse, how is the graph changed? 4. If a and b are equal in the equation of an ellipse, how is the graph changed? 5. If p changes from + to in a vertical parabola, how is the graph changed? 6. If p changes from + to in a horizontal parabola, how is the graph changed?
7 Part 5  *PreAP* Real World Applications 1. A parabolic reflector is in the shape made by revolving an arc of a parabola, starting at the vertex, about the axis of the parabola. If the focus is 9 inches from the vertex, and the parabolic arc is 16 inches deep, how wide is the opening of the reflector?. The face of a onelane tunnel is a square with a semicircle above it. The semicircle has a diameter of 18 ft. A truck that is 15 ft wide and ft tall tries to drive through the tunnel. a. Will the truck fit? b. By how much? (over or under)
8 Conics Units ALL Assignments Part 1 Which conic section is formed by cutting a cone: 1) Diagonally by not cutting through the base ) Parallel to the base 3) If you cut a Double cone Perpendicularly 4) Diagonally through base Write each type of symmetry that the listed conic has: horizontal, vertical, or diagonal 5) Circles 7) Vertical Parabolas 6) Ellipses 8) Horizontal Parabolas Find the center and radius, and then graph the circle. 11) x + y = 81 1) (x  3) + (y 5) = 4 Write the equation of the circle. 15) Center (3, 3) and radius of 4 16) Radius of 5 at center of (1, 1) 13) (x  ) + (y+1) = 16 14) (x  ) + (y 3) = 9 17) 18) 9) Vertical Hyperbolas 10) Horizontal Hyperbolas Part II Answer each question about ellipses. Sketch a picture to support your answer. 1) What happens when a and b are the same? ) What happens when a and b are switched? Find the parts and then graph (see notes). x y 3) 5) (x + 3) (y + ) **PreAP #78 also ) 16x + 9y = 144 6) x + y = 64 8) 16x + 36(y 1) = 576 4) (x 1) (y ) 4 Write the equation. 9) 10) 1 11) Use complete the square to change from General Form to Conic Form 1) x + y + 6x y + 9 = 0 **PreAP#1516 also 13) x + y 16x + 10y + 53 = 0 15) 16x + 9y 18x + 108y = 0 14) x + y + 8x 6y 15 = 0 16) 4x + 9y 48x + 7y = 0 Part III Answer each question about hyperbolas. Sketch a picture to support your answer. 1) What happens when the y and x terms are switched? Find the parts and then graph (see notes). ) (y + 3) (x + 1) = 1 5 (y + 3) 4 3) = 1 64 (x 1) 16 (x ) (y + 5) 4 9 ( x + ) ( y 5) 4 5 4) = 1 5) 6) (x1) + y = 9 **PreAP #7 also 7) 4x y = 4
9 Write the equation. 8) 9) 10) Part IV Answer each question about parabolas. Sketch a picture to support your answer. 1) What happens when you have a horizontal orientation and the p becomes negative? ) What happens when you have a vertical orientation and the p becomes negative? 3) What happens if the x is squared instead of the y being squared? Find the parts and then graph(see notes). 4) (y 3) = 8(x + ) 5) (x ) = 4(y 1) 6) x = 1y 7) y = 8(x + ) 8) x = 16(y ) Use complete the square to change from General Form to Conic Form 11) x  10x  y + 1 = 0 14) y  x + 0y + 49 = 0 1) x + x + y 1 = 0 15) 3x + 30x + y + 79 = 0 13) y  x  8y + 17 = 0 9) ( x 4) ( y ) ) (x + 4) + (y + 1) = 49 16) x + y + x + 8y + 8 = 0 **PreAP only Write the equation. 17) Focus (0,0) directrix y = 4 18) Focus (0,) directrix x = 19) Focus (0,1) directrix x = Part V Determine which conic section each equation represents. 1) y = x ) ( ) x + 5 ( y+) 11) x y x 8 = 0 ) x + y  = 1 y 53 = ) x 1x  y 4y = 1 3) 6x 38x = 0 7) y + 8y = 0 4) (x  ) + (y + 5) = 49 8) x + 3y 13) x + 7y 5x 1y 13 = 0 + 4y + 5x + 6 = 0 5) ( ) x  1 (y+5) 9) 5x 7y + 7x + 9y = ) 9x + y 7x 153 = 0 14) Label each conic section formed by the intersection of a plane with a cone. Answer the questions about parameter changes. 14) What happens when a and b are the same in an ellipse? 15) What happens when a and b are switched in an ellipse? 16) What happens when a and b are switched in a circle? 17) Explain how p affects the graph of a parabola. *PreAP only* Solve the following application problems. 15) How high is a parabolic arch, of span 4ft and height 18ft, at a distance 8ft from the center of the span? 16) The face of a onelane tunnel is a square with a semicircle above it. The semicircle has a diameter of 5m. A truck that is 4.5m wide and 6.5 m tall tries to drive through the tunnel. Will the truck fit?
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