Grade 10

• Conic sections


  Different types: parabola, ellipse, circle, hyperbola, and degenerate cases
• Circle


  A closed curve consisting of all points in a plane that are equidistant from a given point, the center
• Graphing a circle


  1. Determine the standard form of the equation

  2. Graph the circle in a rectangular coordinate system
• Parabola


  A curve that is the set of all points in a plane that are equidistant from a given point, the focus, and a given line, the directrix
• Graphing a parabola


  1. Determine the standard form of the equation

  2. Graph the parabola in a rectangular coordinate system
• Ellipse


  A closed curve consisting of all points in a plane the sum of whose distances from two fixed points, the foci, is constant
• Graphing an ellipse


  1. Determine the standard form of the equation

  2. Graph the ellipse in a rectangular coordinate system
• Hyperbola


  A curve consisting of all points in a plane the difference of whose distances from two fixed points, the foci, is constant
• Graphing a hyperbola


  1. Determine the standard form of the equation

  2. Graph the hyperbola in a rectangular coordinate system
• Recognize the equation and important characteristics of the different types of conic sections
• Systems of nonlinear equations


  Equations that are not linear, involving variables raised to powers other than 1
• Solving systems of nonlinear equations

  1. Substitution

  2. Elimination

  3. Graphing
• Series


  A sequence of numbers or terms that follow a specific pattern
• Sequence


  An ordered list of numbers or terms
• Sigma notation


  A way to represent a series using the summation symbol Σ
• Principle of Mathematical Induction


  A method of proving that a statement is true for all positive integers
• Pascal's Triangle


  A triangular array of numbers in which each number is the sum of the two numbers directly above it
• Binomial Theorem


  A formula for expanding binomial expressions raised to a positive integer power
• Determining a term in the expansion of (x+y)^n

  Without expanding the entire expression
• Conic sections


  Curves formed by the intersection of a plane and a cone
• Types of conic sections

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola
• Circle


  • Formed when the plane is horizontal
• Ellipse


  • Formed when the (tilted) plane intersects only one cone to form a bounded curve
• Parabola


  • Formed when the plane intersects only one cone to form an unbounded curve
• Hyperbola


  • Formed when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyperbola)
• Degenerate conics


  • Point
  • One line
  • Two lines
• Circle


  A special kind of ellipse (when the tilted plane is horizontal)
• Circle


  The collection of all points that are a fixed distance from a given point (the center)
• The distance formula can be used to find the distance between a point and the center of a circle
• Degenerate conics


  A point, one line, and two lines formed when a plane and cones intersect
• Circle


  A special kind of ellipse when the tilted plane is horizontal
• Center of a circle

  The point C from which all points on the circle are equidistant
• Radius of a circle


  The common distance of all points on the circle from the center
• Equation of a circle


  1. PC = r

  2. √(x-h)^2 + (y-k)^2 = r

  3. (x-h)^2 + (y-k)^2 = r^2
• Standard equation of a circle

  • x^2 + y^2 = r^2 (when center is at origin)

  (x-h)^2 + (y-k)^2 = r^2 (when center is at (h,k))
• Circles with given conditions

  • (1) x^2 + y^2 = 16 (center at origin, radius 4)

  (2) (x+4)^2 + (y-3)^2 = 7 (center (-4,3), radius √7)

  (3) (x-3)^2 + (y-1)^2 = 25 (center (3,1), radius 5)

  (4) (x+2)^2 + (y+1)^2 = 16 (center (-2,-1), radius 4)

  (5) (x-3)^2 + (y-2)^2 = 9 (center (3,2), radius 3)

  (6) (x-5)^2 + (y+6)^2 = 25 (center (5,-6), tangent to y-axis)

  (7) (x-5)^2 + (y+6)^2 = 36 (center (5,-6), tangent to x-axis)

  (8) (x-1.5)^2 + (y-3)^2 = 25 (diameter from (-1,4) to (4,2))
• Equations of circles


  • (1) x^2 + y^2 - 6x = 7 (center (3,0), radius 4)

  (2) x^2 + y^2 - 14x + 2y = -14 (center (7,-1), radius 6)

  (3) 16(x+3)^2 + 16(y-5/4)^2 = 484 (center (-3,5/4), radius 11/2)
• Center


  (-4, 3)
• Tangent line


  y = -4x - 30
• Equation of circle


  (x + 4)^2 + (y - 3)^2 = 17