Ex. 1-3: Put each circle below in the correct equation form as listed!! above, then determine the center and radius of each circle.

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1 Day 1 Conics - Circles Equation of a Circle The circle with center (h, k) and radius r is the set of all points (x, y) that satisfies!! (x h) 2 + (y k) 2 = r 2 Ex. 1-3: Put each circle below in the correct equation form as listed!! above, then determine the center and radius of each circle. 1. x2 + y 2 8x + 2y +13 = 0 2. x2 + y 2 + 8x 6y 15 = 0 3. x2 + y 2 +10y 75 = 0

2 Day 2 Conics - Parabolas Day 1 Graph-Translation Theorem - In a sentence for a graph, replacing x with x - h and y with y - k causes the graph to undergo a translation. Consider the parabola y = ax2 as the pre-image. It intersects the origin. The image after a translation of the parabola is:!!!!! y k = a(x h) 2 OR y = a(x h)2 + k a - determines if a parabola opens up or down. It also determines the! shape of the parabola. As the a increases in value, the narrower! the parabola gets, and vice versa. (h, k) - the vertex of the parabola h - translates the parabola left or right k - translates the parabola up or down The axis of symmetry for a parabola that opens up or down is x = h. The maximum or minimum values of a parabola is k. Ex. 1-4: Determine the vertex, axis of symmetry, which way the parabola opens, and the max. or min. value for each parabola. Graph #1 and y = 2(x 1)2 1!!!!! 2. y = 2(x + 3) 2 + 2

3 1.! 2.!!!! 3. y = 3(x + 2)2!!!!!! 4. y = 3x2 + 4 Ex. 5-7: Put each parabola in the form y = a(x h)2 + k. Then find the! vertex, axis of symmetry, which way the parabola opens, and the! max. or min. value of the parabola. 5. y = x2 +10x + 3!!!!!! 6. y = 2x 2 +12x + 20

4 7. y = 3x2 36x 98 Day 3 Conics - Parabolas Day 2 A parabola is a set of all points equidistant from a fixed line called the! directrix(d), and a fixed point not on the line, called the focus(f). The vertex(v) of a parabola is always midway between the focus and the! directrix.! Examples! Up!!!! Down!!! Left!!!! Right

5 A parabola that opens up or down will always have the equation y = ax2. A parabola that opens left or right will always have the equation x = ay2. a is positive for parabolas that open up or right.! a = 1 $ " # 4c% & " a = 1 % a is negative for parabolas that open down or left. # $ 4c& ' c = the distance from the vertex to the focus, or the distance from the! vertex to the directrix. Latus Rectum (LR) - the chord through the focus of a parabola, perpendicular to the axis of symmetry of the parabola, and with endpoints on the parabola. The length of the LR chord is 4c.! -add picture to graph on previous page- For Ex. 1-4, graph and label the vertex, focus, directrix, and LR endpoints of each parabola. Determine the equation of each parabola. 1. Vertex (0, 0)! Focus (0, 7)

6 2. Vertex (0, 0)! Directrix y = 8 3. Vertex (0, 0)! Focus (-6, 0) 4. Vertex (0, 0)! Directrix x = -5

7 Ex. 5-6: Using the given equation, find and graph the vertex, focus,!!!! directrix, and LR endpoints. 5. y = 1 20 x2 6. x = 1 16 y2

8 Day 4 Conics - Parabolas Day 3 Parabolas that open up or down: y = a(x h)2 + k Vertex: (h, k)!!! Axis of symmetry: x = h! Latus Rectum = 4c c = distance from vertex to focus or from vertex to directrix Parabolas that open left or right:! x = a(y k) 2 + h Vertex: (h, k)!!! Axis of symmetry: y = k! Latus Rectum = 4c c = distance from vertex to focus or from vertex to directrix Ex 1-4: Determine the equation of each parabola. If not already given, find the vertex, focus, directrix, axis of symmetry, and LR endpoints. 1. Focus (-3, -3)! Directrix x = -7!

9 2. Vertex (3, -2)! Focus (3, -6)! 3. Focus (-3, 0) Directrix y = 2! 4. Focus (-2, -2) Opens Right Latus Rectum 20!

10 Day 5 Conics - Day 4 Parabolas Ex. 1-2: Put the following equations in the form y = a(x h)2 + k or!! x = a(y k) 2 + h. Determine the vertex, focus, directrix, axis of!! symmetry and LR endpoints for each parabola. 1. x2 4x 12y 8 = 0 2. y2 + 4y + 20x 76 = 0

11 Day 6 Conics - Day 1 Ellipses The equation of an ellipse with center (0, 0) and vertices (a, 0), (-a, 0) x 2! (0, b), and (0, -b) is:! a + y2 2 b = 1 2. Case #1: If a > b 2a = the length of the major axis focal constant sum of focal radii 2b = the length of the minor axis Foci: (c, 0) and (-c, 0) lie on the major axis!! c = a 2 b 2 Case #1: If a < b 2b = the length of the major axis! focal constant! sum of focal radii 2a = the length of the minor axis Foci:! (0, c) and (0, -c)!! lie on the major axis!!!! c = b 2 a 2

12 Ex. 1-2: For each ellipse, find the vertices, foci, and focal constant. 1. x y2 121 = 1!!! Vertices!! Foci!! Focal Constant 2. 36x2 +100y = 0! Vertices!! Foci!! Focal Constant

13 Day 7 Conics - Day 2 Ellipses!! Under a translation of T h,k,!!!!! x 2 a 2 + y2 b 2 = 1 becomes (x h) 2 + a 2 (y k)2 b 2 = 1. A translated ellipse has center (h, k). Ex. 1-2: Find the center, vertices, foci, and focal constant. Graph # x2 + 9y x 36y 164 = 0!! Center!! Vertices!!! Foci!! Focal Constant

14 2. 9x2 + 4y 2 54x + 32y +1 = 0!! Center!! Vertices!!! Foci!! Focal Constant Day 8 Conics - Day 1 Hyperbolas The equation of a hyperbola with center (0, 0) and vertices (a, 0) x 2! and (-a, 0) is:!! a y2 2 b = 1 2 2a =! Transverse axis length (vertex to vertex)!! Focal Constant Difference of Focal Radii Foci: (c, 0) and (-c, 0) where c = a 2 + b 2 Asymptotes:! y = ± b a x

15 The equation of a hyperbola with center (0, 0) and vertices (0, b) y 2! and (0, -b) is:!! b x2 2 a = 1 2 2b =! Transverse axis length (vertex to vertex)!! Focal Constant Difference of Focal Radii Foci: (0, c) and (0, -c) where c = a 2 + b 2 Asymptotes:! y = ± b a x Ex. 1-2: For each hyperbola, find the vertices, foci, focal constant, and!!! asymptotes, then graph x2 25y = 0 Vertices!!!! Foci!! Asymptotes!! Focal Constant

16 2. 9x2 16y = 0 Vertices!!!! Foci!! Asymptotes!! Focal Constant Day 9 Notes - Day 2 Hyperbolas Under a translation of T h,k,!!!!! x 2 a 2 y2 b 2 = 1 becomes (x h) 2 a 2 (y k)2 b 2 = 1. Under a translation of T h,k,!!!!! y 2 b 2 x2 a 2 = 1 becomes (y k) 2 b 2 (x h)2 a 2 = 1.

17 Ex. 1-2: For each hyperbola, find the vertices, foci, focal constant, and!!! asymptotes, then graph x2 9y x + 54y 593 = 0 Center!! Vertices!! Foci!! Focal Constant! Asymptotes 2. 4x2 25y 2 8x 100y + 4 = 0 Center!! Vertices!! Foci!! Focal Constant! Asymptotes

18 Day 10 Conics - Review insert Barry s page here

19 Summarizing Conic Sections Conic!!! Equation with!!!! Equation with!!!! Center at Origin!!! Center NOT at origin Parabola!!! y = ax 2!!!!! y = a(x h) 2 + k!!!! x = ay 2!!!!! x = a(y k) 2 + h Circle!!! x 2 + y 2 = r 2!!! (x h) 2 + (y k) 2 = r 2 Ellipse!!! Hyperbola!!!!!! x 2 a 2 + y2 b 2 = 1!!! x 2 a 2 y2 b 2 = 1!!! y 2 b 2 x2 a 2 = 1!!! If a quadratic equation is given in its general form, (x h) 2 a 2 + (x h) 2 a 2 (y k) 2 b 2 (y k)2 b 2 = 1 (y k)2 b 2 = 1 (x h)2 a 2 = 1!!!!!!! Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, how can you tell which type of conic section it is? The key is to look at! the x2 and y 2 terms - they decide everything. Circle - the coeff. of the x2 and y 2 terms must be the EACT SAME. Ellipse - the coefficients of the x2 and y 2 terms are not the same, but!!! either both positive or both negative. Hyperbola - the coefficients of the x2 and y 2 terms have opposite signs. Parabola - has only an x 2 term only, or a y2 term only

20 Ex. 1-12: Classify each equation (circle, parabola, ellipse, or hyperbola) 1. 3x2 4y 2 + 6x + 8y 7 = 0!! 2. 4x 2 + 4y 2 3x y + 6 = x2 + 2y 2 + 4x + 3y +1 = 0!! 4. x 2 4x +12y 8 = x2 5y 2 5x 5y 10 = 0!! 6. y 2 9x 18y + 27 = 0 7. x2 + y 2 + 3y + 6 = 0!!! 8. x 2 + 3y 2 + 4x + 7 = 0 9. x2 y 2 8x +12 = 0!!! 10. x 2 16y 2 4x + 2 = x2 + y 2 16 = 0!!!! 12. y 2 + 8x + 4y 3 = 0

21 Ex 13-16, if the conic section is:! a) a circle, find its center and radius, then graph.! b) a parabola, find its vertex, focus, directrix, axis of symmetry,!! and latus rectum endpoints, then graph.! c) an ellipse, find its center, vertices, and foci, then graph.! d) a hyperbola, find is center, vertices, and foci, then graph. 13. y2 20x 4y 56 = x2 + y 2 + 4x 6y 68 = 0

22 15. 9x2 + 4y 2 54x + 32y +1 = x2 25y x +150y = 0