Pre calculus 1st sem

• CONIC SECTION is a particular class of curves which often times appear in nature and which have applications in other fields
• Conic sections


  Nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone
• Generating a Circle


  Plane intersects base of cone
• Generating an Ellipse


  Plane intersects cone at an angle
• Generating a Parabola


  Plane intersects cone at an angle to form unbounded curve
• Generating a Hyperbola


  Plane intersects both cones
• Degenerate conic section


  • Generated when a plane intersects the vertex of the cone
  • Types include singular point, line, two intersecting lines
• Circle


  Set of all points in a plane equidistant from a fixed point called the center
• Radius


  Distance from the center of the circle
• Circumference


  Distance around the circle
• Standard Equation of Circle

  C(0,0): 𝑥² + 𝑦² = 𝑟²<|>C(h,k): (𝑥 − 𝑕)² + (𝑦 − 𝑘)² = 𝑟²
• Parabola


  Locus of any point equidistant from a given point (focus) and a given line (directrix)
• Parts of a Parabola

  • Vertex (V)
  • Focus (F)
  • Directrix (d)
  • Axis of Symmetry
  • Latus Rectum
• Vertex (V)


  Minimum or maximum point of the parabola
• Focus (F)


  Point inside of the vertex
• Directrix (d)


  Line opposite of the focus
• Axis of Symmetry


  Line dividing the parabola into mirror images
• Latus Rectum


  Chord passing through the focus, perpendicular to the axis of symmetry
• Standard Equations of Parabola

  y² = 4𝑎𝑥<|>y² = −4𝑎��<|>x² = 4𝑎𝑦<|>x² = −4𝑎𝑦
• Orientation of Parabola


  • Opening to the Right
  • Opening to the Left
  • Opening Upwards
  • Opening Downwards
• Standard Equations of Parabola

  • 𝑥² = 12𝑦
  • 𝑦² = −20𝑥
  • (𝑥 − 10)² = 20(𝑦 + 4)
  • (𝑦 − 3)² = 8(𝑥 − 5)
  • (𝑥 − 16)² = −4(𝑦 − 12)
• Ellipse


  Set of all points in a plane, the sum of whose distances from two fixed points (foci) is a constant
• Circle as a special case of an ellipse


  When a = b
• Parts of an Ellipse

  • Major Axis (a)
  • Minor Axis (b)
  • Center (C)
  • Vertices (V)
  • Co-Vertices (B)
  • Foci (F)
  • Latera Recta (LR)
• Ellipse


  • Semi-major axis
  • Semi-minor axis
  • Center
  • Vertices
  • Co-Vertices
  • Foci
  • Latera Recta
• Semi-major axis


  Half of the major axis
• Semi-minor axis


  Half of the minor axis
• Center


  The intersection of the two axes of symmetry
• Vertices


  The end points of the major axis
• Co-Vertices


  The end points of the minor axis
• Foci


  Always inside the ellipse and contained by the major axis
• Latera Recta


  A chord passing through a focus of an ellipse which is perpendicular to its major axis
• a


  The distance between the Vertices and the Center
• b


  The distance between the Co-Vertices and the Center
• c


  The distance between the Foci and the Center
• Distance between Latera Recta and Center


  𝐛𝟐/𝐚
• Standard Equation of Ellipse with C(0,0)

  • x²/a² + y²/b² = 1, a > b
  • x²/b² + y²/a² = 1, a > b
• Standard Equation of Ellipse with C(h,k)


  • (x − h)²/a² + (y − k)²/b² = 1, a > b
  • (y − k)²/a² − (x − h)²/b² = 1
• Vertices with C(0,0)


  • V1(−a, 0), V2(a, 0)
  • V1(0, a), V2(0, −a)
• Co-Vertices with C(0,0)


  • B1(0, b), B2(0, −b)
  • B1(−b, 0), B2(b, 0)