Conic sections

• Conic sections


  Ellipses, parabolas, hyperbolas, and circles
• Graphing conic sections

  1. Identify equation type (circle, ellipse, parabola, hyperbola)

  2. Put equation in standard form

  3. Plot graph based on standard form
• Standard equation for a circle


  (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius
• Graphing a circle

  1. Identify center (h,k)

  2. Calculate radius r

  3. Plot points at (h±r, k±r) and connect
• Standard equation for an ellipse

  (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center, a is the major axis length, and b is the minor axis length
• Graphing an ellipse

  1. Identify center (h,k)

  2. Determine a and b from equation

  3. Plot points at (h±a, k), (h, k±b) and connect
• Ellipse

  • Uneven, not equal in all sides
  • Major axis is longer than minor axis
• Major axis


  Longer axis of an ellipse
• Minor axis


  Shorter axis of an ellipse
• Vertices

  Endpoints of the major axis
• Foci


  Points along the major axis, inside the ellipse, where c^2 = a^2 - b^2
• Finding ellipse intercepts


  1. Set y=0 to find x-intercepts

  2. Set x=0 to find y-intercepts
• For an ellipse, the length of the major axis is 2a and the length of the minor axis is 2b
• The coordinates of the foci for an ellipse are (h±c, k) where c^2 = a^2 - b^2
• Parabola


  Conic section that is U-shaped
• Hyperbola


  Conic section that is shaped like two intersecting lines
• Finding ellipse properties

  1. Find center

  2. Find a and b

  3. Find foci

  4. Find major vertices

  5. Find major and minor axis lengths
• Finding hyperbola properties

  1. Find center

  2. Find a and b

  3. Find foci

  4. Find vertices

  5. Find asymptotes
• Parabola


  The general equation is y = x^2 if it opens upward, y = -x^2 if it opens downward, x = y^2 if it opens to the right, x = -y^2 if it opens to the left
• Directrix


  The distance between the vertex and the directrix is p
• Focus


  p units above the vertex
• Coordinates of the focus

  p, 0 if the parabola opens to the right or left, 0, p if it opens up or down
• Equation of parabola shifted from origin


  (y - k)^2 = 4p(x - h) if opening right/left, x - h^2 = 4py - k if opening up/down
• Direction parabola opens

  Right if p > 0, left if p < 0, up if p > 0, down if p < 0
• Solving for parabola equation

  1. Identify if it opens right/left or up/down

  2. Find the vertex (h, k)

  3. Find the value of p

  4. Plot the vertex, focus, and directrix

  5. Choose points to graph the parabola
• Graphing parabola y^2 = 8x

  1. Plot vertex at 0, 0

  2. Plot focus 2, 0

  3. Plot directrix x = -2

  4. Choose points like y = 2, 4, -2 to graph the parabola
• Graphing parabola y - 2^2 = 4(x - 3)

  Plot vertex at 3, 2

  2. Plot focus at 4, 2

  3. Plot directrix at x = 2

  4. Choose points like x = 4, 7 to graph the parabola
• Graphing parabola x + 1^2 = -2(y - 3)

  1. Plot vertex at -1, 3

  2. Plot focus at -1.5, 2.5

  3. Plot directrix at y = 7/2

  4. Choose points like y = 1, -1.5 to graph the parabola
• Given 4 equations, identify which is a circle, ellipse, parabola, hyperbola
• Parabola


  Has an x^2 term but not a y^2 term, or vice versa
• Parabola


  Has an x^2 but not a y^2, or has a y^2 but not an x^2
• Hyperbola


  Has a positive x^2 and a -y^2, or vice versa
• Ellipse


  Has positive x^2 and y^2 coefficients
• Circle


  Has equal coefficients for x^2 and y^2
• Putting equations in standard form

  1. Group x's and y's together

  2. Factor out GCF

  3. Complete the square

  4. Divide to get coefficient of 1
• Circle standard form


  • (x - h)^2 + (y - k)^2= r^2
  • Center at h, k
  • Radius is r
• Ellipse standard form


  • (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
  • Center at h, k
  • Major axis 2a, minor axis 2b
  • Foci at h +/- c, k
• Hyperbola standard form


  • (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

  OR

  (y - k)^2 / a^2 - (x - h)^2 / b^2 = 1

  • Center at h, k
  • Foci at h +/- c, k
• Parabola standard form


  • (x - h)^2 = 4p( y - k )

  OR

  ( Y - k)^2 = 4p (x - h)

  • Vertex at h, k
  • Focus at h +/- p, k

  OR

  h, k +/- p
• For hyperbola, c^2 = a^2 + b^2

  For ellipse, c^2 = a^2 - b^2