Definition: Conic Sections The conic sections are all the curves that can be made from slicing two cones stacked tip to tip with a plane.
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1 . The Parabola The main topic of this chapter is something we call conic sections. Definition: Conic Sections The conic sections are all the curves that can be made from slicing two cones stacked tip to tip with a plane. Some tpes of conic sections (or conics) we have alread seen. For eample, if we slice the cones through the tips we get a single point. If we slice the cones in such a wa as we simpl touch one side of one cone and the opposite side of the other, we get a line. Lastl, if we slice them verticall though the tips, we get two intersecting lines. These tpes of conics we have alread dealt with. We are more interested four conic sections demonstrated below. We will start with the parabola since we have alread had some contact with parabolas in chapter 0. Standard Form of a Parabola The equation of a parabola with verte at h, k is a a h k k h for a vertical ais of smmetr of for a horizontal ais of smmetr of h or, k. This gives us the following four possibilities for parabolas, the first two we have seen before.
2 a positive a h k a negative Ais of smmetr verte a positive a k h a negative verte verte Ais of smmetr We can see that graphing these parabolas will be ver similar to how we graphed them in chapter 0. Its just that now its possible for the graph to be sidewas. Eample : Graph the following. State the verte and all intercepts. a. 4 b. 5 Solution: a. First we need to get the parabola into standard form so we can find the verte. We do this b completing the square (recall there is an alternate method of using = -b/a, however, for the purpose of this chapter its best to do use the completing the square method. This wa, all conics can be treated similarl). We proceed as follows
3 4 So our verte is at, Net we need our intercepts. Recall, to find the -intercepts, we set the bottom equation from above to equal 0 and to find the -intercepts we set in the top equation to zero. This gives us -intercepts: -intercept: 0.., Now we simpl plot all of our values and use the fact that the graph opens up (since a = which means a > 0) to complete the graph. b. Similarl we need to start b getting the equation into standard form. So we complete the square on the terms this time to get the following Notice, though, the order here for the verte is reversed. This time we have the verte is, 4. We can alwas remember this b simpl remembering that the value inside the parenthesis alwas goes with the variable inside the parenthesis and that its sign is alwas opposite. The value on the outside alwas goes with the other variable. So here, the goes with the as a and so thus 4 goes with the. Now again we need the intercepts this time the wa we do it is again switched. This time to find the -intercepts, we set the bottom equation from above to equal 0 and to find the -intercepts we set in the top equation to zero.
4 This gives us -intercept: -intercepts: , Now we simpl plot all of our values and use the fact that the graph opens right (since a = which means a > 0) to complete the graph. Eample : Graph the following. State the verte and all intercepts. a. b. Solution: a. Proceeding like in eample we complete the square to find the verte. We get So the verte is at, 8. Net we find the intercepts. -intercepts: -intercept: Graphing we get ,
5 b. Similarl we find the verte. So the verte is at -intercept:,. Net we find the intercepts. -intercepts: i Since the intercepts are comple numbers there must be no -intercepts. So putting it all together and using the fact that the graph opens left (since a < 0) we get. Eercises Graph the following. State the verte and all intercepts
6 Find the equation of the parabola in standard form which satisfies the following. 49. verte,, a, vertical ais of smmetr 50. verte, 4, a, vertical ais of smmetr,,, horizontal ais of smmetr 5. verte 5, runs through the point 5. verte, 5, runs through the point, 5. verte is at the midpoint of, and 9, 8, runs through the point, 8, horizontal ais of smmetr 9, horizontal ais of smmetr 54. verte is at the midpoint of 4, 0 and 5,, runs through the point 4, 0, horizontal ais of smmetr 55. ais of smmetr, verte is at the intersection of the ais of smmetr and the line, runs through the point, 5. ais of smmetr, verte is at the intersection of the ais of smmetr and the line 5, runs through the point,
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