C1 | Mindmap

Simultaneous Equations & Disguised Quadratics
1. Linear
  1. Add/ subtract the two equations to eliminate one variable
  2. Set one equal to y or x and substitute in
2. Quadratic
  1. Substitution
3. Disguised Quadratics
  1. Some equations- eg x^4-x^2-5=0 can be converted to quadratic equations to solve more easily
  2. If there are three orders, ie x^4, x^2, x^0
  3. let A = the middle order, therefore highest order = A^2
Quadratics
1. eg 4x^2+3x+7=0
2. Solving
  1. If the equation factorises
    1. Each bracket = 0
    2. eg.(4x+3)(x-2)=0
      1. x = -3/4 or x = 2
  2. If does not factorise...
    1. Quadratic formula
      1. x= (-b ± √(b^2-4ac) ) / 2a
    2. Complete the square
      1. x^2 ± bx = (x±b/2)^2 - (b/2)^2
        Rearrange to find one or two values of x
      2. a(x+b)^2 + c
        Vertex = (-b, c )
3. Inequalities
  1. If multiplying or dividing by a negative number, REVERSE the sign
  2. Quadratic
    1. Set so that equation = 0
    2. Factorise
    3. If equation > 0 it is where the graph is above the x axis
    4. If equation < 0 it is where the graph is below the x axis
4. Intersections of lines
  1. Set equal to eachother to eliminate y
  2. Remember to get the y values at the end by re-substituting the x values
Gradients, tangents and normals
1. To find a gradient, differentiate the equation and then substitute in the x value
2. The tangent to a curve has the same gradient as the point on the curve it touches
3. y+y-value= m (x + x-value)
4. Stationary points
  1. when dy/dx = 0
  2. solve dy/dx=0 to find stationary points
  3. Differentiate dy/dx to give d^2y/dx^2 . Substitute in x values, if negative then it is a max point, if positive it is a min point
Coordinate Geometry, Lines and Circles
1. Midpoints, gradients and distance between two points
  1. Point A => (x,y) Point B => (w,z)
    1. midpoint = ( (x+w)/2 , (y+z)/2 )
    2. length of the line through AB = √{ (x+w)^2 + (y+z)^2 }
    3. Gradient = (x-w)/(y-z)
2. equation of a line through (a,b) with gradient m is y-b = m(x-a)
3. Circles
  1. Equation of a circle centre (a,b) radius r = (x-a)^2 + (x-b)^2 = r^2
Surds and indices
1. Surds
  1. √m x √n = √mn
  2. √m / √n = √(m/n)
  3. To simplify k/√a multiply by √a / √a
2. Indices
  1. a^(-n) = 1/(a^n)
  2. a^n x a^m = a^(m+n)
  3. a^m / a^n = a^(m-n)
  4. (a^m)^n = a^(m x n)
  5. a^0 = 1
3. a ^ (1/n) = n√a
4. a^ (m/n) = n√a^m
Curve sketching and transformations
1. any graph of the form y=x^n pass through (0,0) and (1,1)
2. y=f(x)
  1. y=f(x) + a is a transformation a units upwards
  2. y=f(x+a) is a transformation -a units to the right
  3. y = f(ax) is a stretch sf 1/a parallel to x axis
  4. y = af(x) is a stretch sf a parallel to y axis

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