Conics Memory aid Jessica Kandis

1. Conics Memory aidJessica Kandis

2. Conics Circle y + x = r 5 r=9 Example : y + x = r 5 + 4= 9 4 Radius the line segment between the center and a point on the circle. are the two points with reference to which ay of the variety of curves is constructed. • Focus are the two points with reference to which any of the variety of curves is constructed. • Major and minor axis • are the same length  Center

3. Conics Ellipse Focus are the two points with reference to which any of the variety of curves is constructed. Vertices: The end points of the major axis Co-vertices: The end points of the minor axis Center Major axis is the longer axis of the ellipse. Minor axis is the shorter axis od the ellipse Semi-major axis is half the length of the longer axis of the ellipse Major axis Semi-minor axis Minor axis Minor axis Major axis Semi- minor axis is half the length of the shorter axis of the ellipse Semi-minor axis Semi-major axis Semi-major axis

4. Formula for an ellipse Length of major axis = 2a Length of minor axis = 2b Sum of focal radii = 2a Length of major axis = 2a Length of minor axis = 2b Sum of focal radii = 2b 5 = 1 4 a= Major axis b= minor axis 4 4 6 6 4 5 Example: Example: = 1 = 1 = 1 = 1

5. Locus of an ellipse L1 L1 L2 L2 When point chosen is on the top to the right, in this case L1<L2 When point chosen is on the bottom to the right, in this case L1<L2 L1 L1 L2 L2 When point chosen is on the bottom to the left, in this case L1>L2 When point chosen is on the top to the left, in this case L1<L2

6. L1 L2 When point chosen is on the top to the right, in this case L1>L2 When point chosen is on the bottom to the right, in this case L1>L2 L2 L1 L2 L1 When point chosen is on the top to the left, in this case L1<L2 When point chosen is on the bottom to the left, in this case L1<L2 L1 L2

7. L1 L1 L2 L2 In these 4 cases, L1=L2 L2 L1 L1 L2

8. Vertices are the points at which a hyperbola makes its sharpest turns. They are on the major axis Hyperbola Foci are the two points next to each curve. Focal radii (L1 and L2) L2-L1 gives a constant number, meaning when choosing any point on the locus, the difference between L2 and L1 will always be the same. Least distance bet curves (transverse axis) L2 L1 Asymptotes are the two lines that that function get closer to but never touches. What helps to find the conjugate axis Conjugate axis

9. Formula for the hyperbola x - y y– x = 1 =-1 OR ab b a 3 Example : 5 5 - 0 0 – 5 = 1 =-1 OR 53 35 Chosen point (5,0)

10. Hyperbola Other way of seeing a hyperbola. They have the same properties Vertices are the points at which a hyperbola makes its sharpest turns. They are on the major axis L1 Foci are the two points next to each curve. Focal radii ( L2 and L1 ) L2-L1 gives a constant number, meaning when choosing any point on the locus, the difference between L2 and L1 will always be the same. L2 Least distance bet curves (transverse axis) Asymptotes are the two lines that that function get closer to but never touches. What helps to find the conjugate axis Conjugate axis

11. Formula for the hyperbola y - x x - y = 1 =-1 OR ba a b 3 Example: -3 - 0 0 - -3 4 = 1 =-1 OR 3 4 43 Chosen point (0.-3)

12. Parabolas y x x y x y x y

13. Vertex (h,k) 1 -4 3 6

14. 4 -7 1 6 The h and k make up the vertex.

15. Parabola Focus and Directrix Focus (0,c) Focus (0,-c) Directrix y=-c Directrix y=c Focus is a point on the interior of the parabola used in the formal definition of the curve Directrix is a line perpendicular to the axis of symmetry Focus (c,0) Focus (-c,0) Directrix x=c Directrix x=-c

16. L1 L2 90 degree angle The distance the Focus and any point on the parabola will ALWAYS be the same measurement as the distance from a point from the directrix making a 90 degree angle, to the same point on the parabola. These 2 lines are called the locus (L1 and L2). In conclusion… L1=L2

17. More examples L2 L2 L1 L1 L2 L1

18. C= distance between the focus and the vertex C= distance between the vertex and the directrix If already given the c and you need to find a… c=5 Rule: Example: So the c would be c=|1/4(0.5)|  C=|0.5| a = 0.5

19. The basic info: • all 4 conics The ones in color are the ones that I’ve done • their locus definitionsThere are probably more that I’ve done that I haven’t • their graphs, in all their possible orientationsput in color but here are already 15 points that I have • the rules that correspond to the conics/graphsalready done  • the parameters for each conic: where they are in the rule, where they are in the graph • all the parts of the conics, names, definitions, locations • any and all formulas pertaining to parts • any and all properties of the conic • Organization: • information presented in an orderly, logical, easy-to-follow way • visual information as well as textual info, AND correspondence between them • colour coding that makes it easy to see the organization – ie the parameter a is always coloured red in text and graph, or all your ellipse info is turquoise…. • Depth: • the information is organized in your own order, so that I can see you didn’t just copy down all the notes I gave you • comparisons between conics • something new – something you noticed but that I never said • Rigour: • worked-out examples, like finding the rule given certain info, or solving your own problem involving conics • actual concrete drawings of conics done with string(s) (hey, you could video yourself like I did! But no walls.) (I’m serious, put the pen down.) • alternate forms of the rule • inequations involving conics • Extra: • any use of tech that enhances the message (codecogs, geogebra demos, a voicethread, embedding using scribd……) • publishing on blog without handwriting