Core 1 Key points

• Simplifying expressions --> Like terms & rules of indices 
  • A^m x A^n = A^m+n
  • A^m / A^n = A^m-n
  • A^-m = 1/A^m
  • A^1/m = m√A
  • A^n/m = m√A^n
  • (A^m)^n = A^mn
  • A^0 = 1
• Expanding expressions --> Each term inside X each term outside --> Factorising = opposite
• Quadratic expression --> ax^2 + bx + c
• Difference of squares --> x^2 - y^2 = (x+y)(x-y)
• Surd = √prime number
• Manipulation surds
  • √ab = √a x √b
  • √a/b = √a / √b
• Rationalise surds
  • 1/√a --> multiply top and bottom by √a
  • 1/(a+√b) --> multiply top and bottom by (a-√b) --> opposite for 1/(a-√b)

• General form --> y=ax^2 + bx + c
• Solving quadratic equations
  • factorisation
  • Completing the square --> x^2 +bx = (x + b/2)^2 - (b/2)^2
  • Formula --> x= -b+-√(b^2 - 4ac) / 2a
• QE has 2 solutions - may be =
• To sketch a Quadratic graph 
  • Shape: a > 0 = Happy , a < 0 = Sad
  • X-axis and y-axis crossing points
  • discriminant b^2 - 4ac --> general shape

• Linear simultaneous equations --> elimination or substitution
• 1. linear and 1. quadratic (simultaneous) --> substitutiom
• X or / inequality by -ve no. --> </> = opposite
• To solve a quadratic inequality
  • Solve corresponding quadratic equation
  • sketch graph of quadratic function
  • use sketch to find required set of values.

• Shapes of the basic curves

  y = x^2           y = x^3      y = (x-a)(x-b)(x-c)     y=1/x

• Transformation
  • f(x+a) --> -a in the x-direction
  • f(x) + a --> +a in the y-direction
  • f(ax) --> stretch of 1/a in x-direction (X x-coordinates by 1/a)
  • af(x) --> stretch of a in y-direction (X y-coordinate by a)

• General form - y = mx + c --> m = gradient & c = y intercept
• Gradient of line joining two points together
  • m = y2-y1 / x2 - x1
• equation of line w/ m and passes through (x1, y1)
  • y - y1 = m(x -x1)
• equation of line passing through two points
  • y-y1/y2-y1 = x-x1/x2-x1
• perpendicular has gradient -1/m
• two lines perpindicular gradient = -1.

• sequence = series of numbers with a set rule
• each number = term
• nth term = general term
• expressed as a formular for the nth term e.g. Un = 4n + 1
• Arithmetic sequence
  •  Uk+1 = Uk + n
  • a + (a+d) + (a + 2d) + (a + 3d) + (a + 4d)
• nth term of an arithmetic seqence 
  • a + (n-1)d --> a = 1st term and d = common difference
• Sum of an arithmetic sequence
  • Sn = n/2(2a + (n-1) d)
  • Sn = n/2(a+L) --> L = last term
• Use ∑ 
  • 10∑r=1 (5 + 2r) = 7 + 9 + ... + 25

• m of y = f(x) @ specific point = m of tangent to curve @ that point
• m  of tangent @ any point = dy/dx
• f'(x) = derived function
• y = f(x) --> dy/dx = y = f'(x) --> dy/dx = nx^n-1
• 2nd order derivative = d^2y/dx^2
• Equation of tangent to curve @ point A
  • y - f(a) = f'(a)(x-a)
• Equation of normal to curve @ point A
  • y - f(a) = -1/f'(a) (x-a)

• If dy/dx = x^n --> y = (1/n+1)(x^n+1)+c
• If dy/dx = kx^n --> y = (kx^n+1/n+1) + c
• ∫x^ndx = (x^n+1/n+1) +c