Pure

• Implication- A⇒B


  A is sufficient for B (if A is true then B must be true), number ends in 5⇒5 is a factor
• Implication- A⇐B


  A is necessary for B (needed to be true but doesn't guarantee truth), number is even⇐4 is a factor
• Implication- A⇔B


  A&B are both necessary & sufficient for each other, 10 is a factor⇔ends in 0
• Proofs- Deduction


  Logical argument using algebra that shows it must be true
• Proofs- exhaustion


  Trying all possible cases
• Proofs- Counter-example


  Giving a counter-example to prove that it can't be true
• Quadratic inequalities

  Ensure that you check the inequality sign to see if its outside or inside the inequality values
• Discriminants-

  If b2-4ac > 0 it has 2 real roots

  If b2-4ac = 0 it has 1 real (repeated) root

  If b2-4ac < 0 it has no real roots
• Equation of a line-


  With known gradient & coordinate (x1, y1)- y-y1=m(x-x1) 
• Equation of a circle

  centre (a,b), radius=r-(x-a)^2+(y-b)^2=r^2
• Transformations- f(x)+a

  Translate up by a
• Transformations- f(x)-a

  Translate down by a
• Transformations- f(x-a)

  Translate right by a
• Transformations- f(x+a)

  Translate left by a
• Transformations- f(x-a)+b

  Translation by (a,b)
• Transformations- -f(x)

  Reflection in the x-axis
• Transformations- f(-x)

  Reflection in the y-axis
• Transformations- af(x)

  Vertically stretched by a (enlarged)
• Transformations- f((1/a)x)

  Horizontally stretched by a
• Transformations- f(ax)

  Horizontally stretched by 1/a
• Transformations- -af(X)

  Reflection & vertical stretch by a
• Set notation-

  Use curly brackets, x with colon before inequality, use and & or symbols

  E.g.: {x: x>-2}∩{x: x<3}, {x: x<5}∪{x: x>7}
• When calculating the equation from 3 given points on the circumference-


  Calculate the gradient of each of their chords, find the equation of their perpendicular bisectors, use simultaneous to find the centre (intersection), calculate the radius as the line from centre to one of the points on the circumference, substitute into general formula.
• Trig values- sinθ-

  • Repeats every 360
  • sinθ=cos(90-θ)
  • sinθ=sin(180-θ)
  • sinθ=-sin(-θ)
  • sinθ=(360+θ)
• Trig values- cosθ-

  • Repeats every 360
  • cosθ=sin(θ+90)
  • cosθ=cos(360-θ)
  • cosθ=cos(-θ)
  • cosθ=cos(360+θ)
• Trig values- tanθ-

  • Repeats every 180
  • tanθ=-tan(-θ)
  • tanθ=tan(180+θ)
• Differentiation-


  If y=x^n then dy/dx = nx^n-1

  Used to calculate the gradient
• Maximum point-


  $d^2y/dx^2$ < 0
• Minimum point-


  $d^2y/dx^2$ > 0
• Integration-


  Reverse of differentiation

  Integral of x^n = (x^n+1)/n+1 +c

  Remember to add constant when indefinite or substitute when definite
• Area under a curve-


  Definite integral of the curve's function where the limits is the x-values you're calculating the area between
• Vectors-


  ai + bj

  a=distance in the x-direction

  b=distance in the y-direction

  i & j = constants equivalent to 1 in respective direction
• Log rules-

  • if a^x = b then log a b = x
  • log xy = log x + log y
  • log x/y = log x - log y
  • log 1 = 0
  • log x^n = n log x
  • log 1/y = -log y
  • log n√x = log x^1/n = 1/n log x
• Natural logs-

  • e^x = log e x = ln x
  • ln e = 1
• Uses of logs-


  Can prove exponential relationships by making the graphs linear
• Gradient of a tangent to a curve


  Gradient at that point on the curve