pre-calculus

• Conic sections


  The curves which can be derived from taking slices of a "double-napped" cone
• Conic


  A shape generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant
• Forming conic sections


  1. When the plane figure cuts the double napped cone horizontally, it forms a circle

  2. When the plane figure is tilted and cuts only one cone to form a bounded curve, it generates an ellipse

  3. When the plane figure cuts the double napped cone not necessarily vertical to form two unbounded curves, it generates a hyperbola
• Circle


  Set of all points in a plane with fixed distance called the radius from a fixed point called the center
• Standard equation of a circle

  (x - h)^2 + (y - k)^2 = r^2
• Circles


  • Center at (0,0), radius 4
  • Center at (-4, 3), radius √7
  • Center at (5, -6), tangent to the y-axis
• General equation of a circle

  Ax^2 + Ay^2 + Cx + Dy + E = 0
• General equation of a circle

  • x^2 + y^2 - 6x - 7 = 0
  • x^2 + y^2 - 14x + 2y + 14 = 0
• Parabola


  Set of all points in a plane such that the distance from a point to a focus is equal from the same point and the directrix
• Standard equation of the Parabola

  Upward: x^2 = 4ay

  Downward: x^2 = -4ay

  Right: y^2 = 4ax

  Left: y^2 = -4ax
• Parts of a Parabola

  • Vertex
  • Focus
  • Latus rectum
  • Axis of symmetry
  • Directrix
• Vertex


  • Sharpest turn point of parabola
• Focus


  • A point which is used to determine or define the parabola. Distance of the focus to the vertex is determined by variable a/p in the formula.
• Latus rectum


  • Line passing through the focus. Distance of latus rectum is determined by 4a/4p
• Axis of symmetry


  • Line that divides parabola in half
• Directrix


  • Line perpendicular to axis of symmetry, found below the parabola
• Ellipse


  The set of all points in a plane such that the sum of the distances from two points (foci) is a constant
• Chord


  Segment with endpoints on the ellipse
• Major Axis


  Chord lying on the focal axis
• Minor Axis


  Chord through the center, perpendicular to the focal axis
• Semi-major axis


  The number a
• Semi-minor axis


  The number b
• Standard equation of an Ellipse

  Horizontal: x^2/a^2 + y^2/b^2 = 1

  Vertical: x^2/b^2 + y^2/a^2 = 1
• Ellipses


  • Center at the origin

  Center at (h,k)
• Hyperbola


  A set of points in the plane such that the difference of the distances from two fixed points, called foci, remains constant
• Standard equation of a Hyperbola

  Horizontal: x^2/a^2 - y^2/b^2 = 1

  Vertical: x^2/b^2 - y^2/a^2 = 1
• Hyperbolas


  • Center at the origin (0,0)

  Center at (h, k)
• Eccentricity


  A ratio of the distance from the focus and the distance from the directrix.

  Circle: e = 0

  Ellipse: 0 < e < 1

  Parabola: e = 1

  Hyperbola: e > 1
• Conic sections have applications in problem solving