2 9.1 THE ELLIPSE Objectives Graph ellipses centered at the origin Write equations of ellipses in standard formGraph ellipses not centered at the originSolve applied problems involving ellipses
3 Definition of an ellipse All points in a plane the sum of whose distances from 2 fixed points (foci) is constant.If an ellipse has a center at the origin and the horizontal axis is 2a (distance from center to right end is a) and the vertical axis is 2b (distance from the center to the top is b), the equation of the ellipse is:
4 GraphCenter is at the origin, horizontal axis=6 (left endpt (-3,0), right endpt (3,0), vertical axis = 8 (top endpt (0,4), bottom endpt (0,-4))
5 What is c?c is the distance from the center to the focal point
6 What is the equation of an ellipse, centered at the origin with a horizontal axis=10 and vertical axis=8?
7 What if the ellipse is not centered at the origin? If it is centered at any point, (h,k), the ellipse is translated. It is moved right “h” units and up “k” units from the origin.Consider:The center is at (2,-3), the distance from the center to the right & left endpt = 2, the distance to the top & bottom endpt = 1Graph on next slideSince a>b, the horizontal axis will be the major axis and the focal points will be on that axis of the ellipse.
9 What is the distance from the center to each focal point What is the distance from the center to each focal point?(c) If the center is at (2,-3), the foci are at Since the major axis is the horizontal one, you move c units left & right of the center.
10 9.2 The Hyperbola Objectives Locate a hyperbola’s vertices & foci Write equations of hyperbolas in standard formGraph hyperbolas centered at the originGraph hyperbolas not centered at the originSolve applied problems involving hyperbolas
11 Definition of a hyperbola The set of all points in a plane such that the difference of the distances to 2 fixed points (foci) is constant.Standard form of a hyperbola centered at the origin:Opens left & rightOpens up & down
12 What do a & b represent?a is the distance to the vertices of the hyperbola from the center (along the transverse axis)b is the distance from the center along the non-transverse axis that determines the spread of the hyperbola (Make a rectangle around the center, 2a x 2b, and draw 2 diagonals through the box. The 2 diagonals form the oblique asymptotes for the hyperbola.)
13 Focal PointsThe foci (focal points) are located “inside” the 2 branches of the hyperbola.The distance from the center of the hyperbola to the focal point is “c”.Move “c” units along the transverse axis (vertical or horizontal) to locate the foci.The transverse axis does NOT depend on the magnitude of a & b (as with the ellipse), rather as to which term is positive.
14 Describe the ellipse: 1) opens horizontal, vertices at (4,0),(-4,0) 2) opens vertically, vertices at (0,5),(0,-5)3) opens vertically, vertices at (0,4),(0,-4)4) opens horizontal, vertices at (5,0),(-5,0)
15 What if the hyperbola is not centered at the origin? (translated) A hyperbola with a horizontal transverse axis, centered at (h,k) is of the form:
16 Describe the hyperbola & graph Transverse axis is verticalCentered at (-1,3)Distance to vertices from center= 2 units (up & down) (-1,5) & (-1,1)Asymptotes pass through the (-1,3) with slopes = 2/3, -2/3Foci units up & down from the center ,
18 9.3 The Parabola Objectives Graph parabolas with vertices at the originWrite equations of parabolas in standard formGraph parabolas with vertices not at the originSolve applied problems involving parabolas
19 Definition of a parabola Set of all points in a plane equidistant from a fixed line (directrix) and fixed point (focus), that is not on the line.Recall, we have previously worked with parabolas. The graph of a quadratic equation is that of a parabola.
20 Standard form of a parabola centered at the origin, p = distance from the center to the focus Opens left (p<0),or right (p>0)Opens up (p>0) or down (p<0)Distance from vertix to directrix = -p
21 Graph and describe Write in standard form: (1/2)y = (4p)y, thus ½ = 4p, p = 1/8Center (0,0), opens up, focus at (0,1/8)Directrix: y = -1/8
22 Translate the parabola: center at (h,k) Vertical axis of symmetryHorizontal axis of symmetry
23 If the equation is not in standard form, you may need to complete the square to achieve standard form.Find the vertex, focus, directrix & graphVertex (-2,-1), p= -2, focus: (-2,-3), directrix: y=1Graph, next slide
25 9.4 Rotation of Axes Objectives Identify conics without completing the squareUse rotation of axes formulasWrite equations of rotated conics in standard formIdentify conics without rotating axes
26 Identifying a conic without completing the square (A,C not equal zero) Circle, if A=CParabola, if AC=0Ellipse if AC Not equal 0, AC>0Hyperbola, if AC<0
27 A Rotated Conic Section Can the graph of a conic be rotated from the standard xy-coordinate system?YES! How do we know when we have a rotation? When there is an xy term in the general equation of a conic:
28 Rotation of Axes A conic could be rotated through an angle The xy-coordinate system is the standard coordinate system. The x’y’-coordinate system is the rotated system (turning the rotated conic into the standard system)Coordinates between (x,y) and (x’,y’) for every point are found according to this relationship:
29 Expressing equation in standard form, given a rotated axis. Given the equations relating (x,y) and (x’,y’), find (x’,y’) given the angle of rotationSubstitute these expressions in for x in the equation of the rotated conic. The result is an equation (in terms of x’ & y’) that exists IF the equation were in standard position.
30 How do we determine the amount of rotation of the axes?
31 Identifying a conic section w/o a rotation of axes
32 9.5 Parametric Equations Objectives Use point plotting to graph plane curves described by parametric equationsEliminate the parameterFind parametric equations for functionsUnderstand the advantages of parametric representations
33 Plane Curves & Parametric Equations Parametric equations: x & y are defined in terms of a 3rd variable, t: f(t)=x, g(t)=yVarious values can be substituted in for t to produce new values for x & yThese values can be plotted on the xy-coordinate system to generate a graph of the functions
34 Given a function in terms of x & y, can you find its representation as parametric equations? Begin by allowing one variable (usually x) to designated as t. Replace x with t in the expression. Now y is stated in terms of t.x may be replace with other expressions involving t. The only restriction is that x and the new expression for x (in terms of t) must have the same domain.
35 9.6 Conic Sections in Polar Coordinates ObjectivesDefine conics in terms of a focus and a directrixGraph the polar equations of conics
36 Focus-Directrix Definitions of the Conic Sections
37 Polar Equations of Conics (r,theta) is a point on the graph of the conice is the eccentricityp is the distance between the focus & the directrix