Core 1

Core 1

Chapter 1 ~ Algebra and Functions
1. Like terms
2. Multiplying brackets
3. Rules of Indices
4. Factorising
5. Surds
  1. √(ab)=√(a)x√(b)
  2. √(a/b)=√(a)/√(b)
  3. Rationalising
    1. Multiply top and bottom by denominator with opposite central sign
Chapter 2 ~ Quadratic Functions
1. y = f(x) = ax^2 + bx +c
2. Solve quadratic equations by factorisation
3. Completing the square
  1. 1) Write formula > y=x^2 + bx + c
    1. 2) Make a gap appear > y=x^2 + bx ________________ + c
      1. 3) In the gap add on, then take off (0.5*b)^2 > y=x^2 + bx + (0.5*b)^2 - (0.5*b)^2 + c
        4) Add brackets > y=(x^2 + bx + (0.5*b)^2) - (0.5*b)^2 + c
        5) Factorise large bracket and tidy up numbers at the end
4. x = [-b±√((b^2)-4ac))]/2a
5. Discriminant
  1. b^2>4ac and a>0
    1. Two different roots
    2. u shape crosses x-axis twice
  2. b^2=4ac and a>0
    1. Two equal roots
    2. u shape sits on x-axis
  3. b^2<4ac and a>0
    1. No real roots
    2. u shape that doesn't touch x-axis
  4. b^2>4ac and a<0
    1. Two real roots
    2. n shape crosses x-axis twice
  5. b^2=4ac and a<0
    1. Two equal roots
    2. n shape sits on x-axis
  6. b^2<4ac and a<0
    1. No real roots
    2. n shape that doesn't touch the x-axis
Chapter 3 ~ Equations and Inequalities
1. Solving simultaneous linear equations
  1. Elimination
  2. Substitution
2. Solving simultaneous equations where one quadratic and one linear equation
  1. Substitution
3. Inequalities
  1. Solve similar to equations
  2. Number lines
  3. Sketches
Chapter 4 ~ Sketching Curves
1. Cubic Curves
  1. y = ax^3 + bx^2 + cx + d
  2. Use factors of equation to work out where the curve crosses the x-axis
    1. x-axis intersections are when x = 0
      1. eg (x-2) > intersection at (2,0). (x+5) > intersection at (-5,0)
  3. y = x^3
    1. Smooth curve through (0,0)
2. Reciprocals
  1. y = k/x
    1. When k>0, curves appear in quadrants where both values are either positive or negative
    2. When k<0, curves appear in quadrants where one value is positive and the other is negative
  2. The further away k is from 0, the further away the curves are from the axes
3. Transformations
  1. f(x+a) > moves whole curve -a in the x-direction
  2. f(x)+a > moves whole curve +a in y-direction
  3. f(ax) > multiply x-coordinates by (1/a)
  4. af(x) > multiply y-coordinates by a
Chapter 5 ~ Coordinate Geometry in the (x,y) Plane
1. y = mx + c
  1. m is the gradient and c is the y-intercept
2. ax + by +c = 0
  1. a, b, and c are all integers
3. Gradient between two points = (y2 - y1)/(x2 - x1)
  1. Equation of a line using one point and the gradient > y - y1 = m(x - x1)
  2. Equation of a line between two points > (y - y1)/(y2 - y1) = (x-x1)/(x2 - x1)
4. Two lines
  1. Perpendicular
    1. Gradient = -1/m
    2. The product of two perpendicular lines is -1
  2. Parallel
    1. Same gradient
Chapter 6 ~ Sequences and Series
1. General term > nth term
  1. a + (n-1)d
2. Un = 4n + 1
3. C1 only has arithmetic sequences
  1. a > first term
  2. d > common difference
4. Sum of an arithmetic sequence
  1. Sn = (n/2)[2a + (n - 1)d]
  2. sn = (n/2)(a + L)
    1. L is the last term
  3. ∑_(r=1)^10(5+2r) =7+9+...+25
Chapter 7 ~ Differentiation
1. Used to work out the gradient of a tangent
2. f(x) = x^n
  1. f'(x) = nx^(n-1)
  2. To get f'(x), multiply power by number in front of x, then reduce power by 1 for each part separately
3. (d^2y)/(dx^2) = f''(x)
Chapter 8 ~ Intergration
1. If dy/dx = x^n, the y = (1/(n+1))(x^(n+1)) + c
2. ∫x^n dx= x^(n+1)/(n+1)+c
3. Calculate c when given any point that the function of the curve passes through
4. Reverse of differentiation

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