This next unit is called conic sections and deals with the equations of parabolas, ellipses and hyperbolas. It has a number of formulas in it that are very similar so it's easy to mix them up. Pretty much, the problems in this section consist of two main problem types: 1) getting the formula into the standard form for an ellipse, hyperbola or parabola 2) Given some information about the conic section, use that information to figure out the equation Think back to the equations of a line. You know the formula is y=mx+b, but if you are not given m and b, you can use information given such as two points or the slope and a point to find the needed information m and b. Then you can fill out the m and b to write the line's equation. The idea is similar here except the equations are a little more involved and often require more algebraic manipulation to get the equation to standard form. Attached below are the notes, and answer keys for the home works as well as links to some good videos explaining in more detail. Look through the work and let me know what questions you have if you get stuck on something. This post will contain some more explanations on parabolas and ellipses. A second post will be out soon discussing hyperbolas in more detail. The formulas that you should have for handy reference are on pages 745, 747, 755, 757, 763, and 776 (776 deals with horizontal and vertical shifts of the different conic sections). First Section: Parabolas The first conic section to be discussed will be the parabola. The two parts that define the path of a parabola are the Focus ( a point) and the directrix (a line). The definition of a parabola is the set of all points equally distant from a focus and the directrix. If some point (x,y) on a given parabola is a distance of 5 from the focus, it is also 5 from the directrix. If another point(x2,y2) is a distance of 10 from the focus on the this same parabola, it is also a distance of 10 from the directrix. Every point on the parabola meets this requirement. The focus will be somewhere on the axis of symmetry of the parabola (the line that bisects the parabola). The directrix line and the focus point are on the opposite sides of the turning point or vertex of the parabola and the vertex will be halfway between the focus and directrix. General format of equations: Given a parabola with a vertical axis of symmetry. (points up and down) and a vertex at (0,0) Vertex = (0, 0) Focus point is (0,P) (P = focal point, is on the y axis) Directrix is the equation y = - p General format of the equation is: x^2 = 4py Given a parabola with a horizontal axis of symmetry (points left and right) and a vertex at (0,0) Vertex = (0,0) Focus point is (P,0) (P = focal point on the x axis) Directrix is the equation x = -p General format of the equation is y^2 = 4px If the parabola has been shifted either horizontally, vertically or both, the general format of the equations are: vertical parabola = ( x-h)^2 = 4p(y-k) where the vertex has been shifted to the point (h, k) Note how the sign of h and k change when going between equation and vertex. horizontal parabola = (y - k)^2 = 4p(x-h) where the vertex has been shifted to the point (h,k) Note in the formulas above that: The x coordinate of the vertex, (h) is always with the x in the equation The y coordinate of the vertex ,(k) is always with the y in the equation. When trying to find the vertex, focus and directrix of a parabola that does not have its vertex at the origin (0,0), it is often helps to figure out what these would be without the shift and then applying the shift. For example: Equation = (x-2)^2 = 12(y-3) This is the formula for a parabola that is vertical The parenthesis tell us the shift is 2 right on the x axis and 3 up on the y (Shift is always in the opposite direction of the sign) Shift = (2,3) Totally ignore the shift in parenthesis and the equation looks like x^2 = 12 y This is the general form of x^2 = 4p y 1) Solve for p. 4p = 12 in this example (The coefficient in front of y always = 4p). So p must = 3. Therefore from the formulas above: Pre - shift Shift(2,3) Post Shift Vertex = (0,0) x+2, y+3 Vertex = (2,3) Focal point = (0,3) Focal Point = (2,6) Directrix = y= - 3 Directrix y = 0 (added 3 to the y from the shift) Attached below are notes explaining in more detail and examples worked out as well as some helpful links. The answers to the homework questions are posted again as well. Homework problems: p. 751 #'s 2,4,6,12,28,40 pp. 781-782 #'s 6,8,14,22 General description of various conic sections. www.youtube.com/watch?v=iJOcn9C9y4w Description of parabola conic section and basic equations www.youtube.com/watch?v=k7wSPisQQYs Parabola part II www.youtube.com/watch?v=CKepZr52G6Y There are a number of specific examples on this page for graphing parabolas with and without shifts. Once you click on the link below, click Ctrl f and a search box opens up. Type parabola and you will be taken to the section of the page with many different videos. www.mathispower4u.com/alg-2.php Section on ellipses Geometric definition of an ellipse - The set of all points in a plane whose sum of distances from 2 fixed points called foci (plural of focus) is constant. The shape of an ellipse is like a circle that got squashed so it resembles a football. Inside the ellipse are two fixed points called foci. Pick any point on the ellipse and find the distance from it to focus 1, call it d1. From that same point, find the distance from it to focus 2, call it d2. The sum of d1 + d2 will always add to the same number no matter what point you chose to pick. Just like with the parabolas, these problems rely on knowing the general format of the equations and applying them based on the given information. x^2/a^2 + y^2/b^2 = 1. Equation of an ellipse centered at the origin. The general formats for the equations of ellipses centered at the origin and with shifts are in the word document below. The orientation of the ellipse, whether elongated more along the vertical axis or the horizontal axis is determined by which is the bigger denominator, a^2 or b^2. If the bigger denominator is under the x, then the main axis (longest) is the horizontal. If the biggest denominator is under the y, then the main axis (longest) is the vertical. So when graphing, the important thing is which coordinate is over the larger denominator, x or y. Below are some links showing the basic graph of an ellipse, and to solve some problems. Also included are notes with problems worked out and the answer key with all the homework problems worked out in detail. As always, please email me with any specific questions. Some time by the end of this week or early next week, you will have a short assessment on these problems. Homework problems: pages 759-60: 2, 4, 6, 10, 14, 20, 30 pages 781-82: 2, 4, 16 ( Hint for when you have difficulty getting the coefficient in front of the x^2 or y^2 to 1 when the right side is already 1 e:g: 9x^2 / 1 + 10y^2 / 1 = 1) Change the coefficient in front of the x^2 or y^2 to 1 and represent the denominator as 1/Coefficient of what had been in front of the x or y. The above equation is equal to: x^2 / (1/9) + y^2 / 1/10 = 1 This is now in standard form. Link for ellipses. Basic concepts. Parts 1 and 2 www.youtube.com/watch?v=LVumLCx3fQo www.youtube.com/watch?v=oZB69DY0q9A Graphing ellipses centered at the origin with major axis along x and along y www.youtube.com/watch?v=azI5kALyiXs&feature=youtu.be (horizontal axis) www.youtube.com/watch?v=3qckea8OuN8&feature=youtu.be (major axis vertical) Graphs not centered at the origin: www.youtube.com/watch?v=tJmp1PJD9o8&feature=youtu.be www.youtube.com/watch?v=oWGyVpq94CM&feature=youtu.be Given an equation, rewrite in standard form and graph. www.youtube.com/watch?v=-i48L0WQ-2I&feature=youtu.be www.youtube.com/watch?v=CL7SNu-5riw&feature=youtu.be Given some information about the ellipse (e.g focus, vertex, etc), find the equation. www.youtube.com/watch?v=k5gMLDjMfvI&feature=youtu.be www.youtube.com/watch?v=RWaEIJOlHlw&feature=youtu.be There are a number of videos on this website for ellipses: www.mathispower4u.com/alg-2.php in addition to the ones posted above. As far as a timeline goes, I would aim to have the questions in this post done by Monday of next week - 4/27/20. As always, email me with any questions
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