C1

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A Levels Maths Mind Map on C1, created by luisnorth on 25/02/2014.
luisnorth
Mind Map by luisnorth, updated more than 1 year ago
luisnorth
Created by luisnorth over 10 years ago
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Resource summary

C1
  1. Simultaneous Equations & Disguised Quadratics
    1. Linear
      1. Add/ subtract the two equations to eliminate one variable
        1. Set one equal to y or x and substitute in
        2. Quadratic
          1. Substitution
          2. Disguised Quadratics
            1. Some equations- eg x^4-x^2-5=0 can be converted to quadratic equations to solve more easily
              1. If there are three orders, ie x^4, x^2, x^0
                1. let A = the middle order, therefore highest order = A^2
              2. Quadratics
                1. eg 4x^2+3x+7=0
                  1. Solving
                    1. If the equation factorises
                      1. Each bracket = 0
                        1. 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
                                1. Rearrange to find one or two values of x
                                2. a(x+b)^2 + c
                                  1. Vertex = (-b, c )
                            3. Inequalities
                              1. If multiplying or dividing by a negative number, REVERSE the sign
                                1. Quadratic
                                  1. Set so that equation = 0
                                    1. Factorise
                                      1. If equation > 0 it is where the graph is above the x axis
                                        1. If equation < 0 it is where the graph is below the x axis
                                      2. Intersections of lines
                                        1. Set equal to eachother to eliminate y
                                          1. Remember to get the y values at the end by re-substituting the x values
                                        2. Gradients, tangents and normals
                                          1. To find a gradient, differentiate the equation and then substitute in the x value
                                            1. The tangent to a curve has the same gradient as the point on the curve it touches
                                              1. y+y-value= m (x + x-value)
                                                1. Stationary points
                                                  1. when dy/dx = 0
                                                    1. solve dy/dx=0 to find stationary points
                                                      1. 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
                                                    2. 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 )
                                                            1. length of the line through AB = √{ (x+w)^2 + (y+z)^2 }
                                                              1. Gradient = (x-w)/(y-z)
                                                            2. equation of a line through (a,b) with gradient m is y-b = m(x-a)
                                                              1. Circles
                                                                1. Equation of a circle centre (a,b) radius r = (x-a)^2 + (x-b)^2 = r^2
                                                              2. Surds and indices
                                                                1. Surds
                                                                  1. √m x √n = √mn
                                                                    1. √m / √n = √(m/n)
                                                                      1. To simplify k/√a multiply by √a / √a
                                                                      2. Indices
                                                                        1. a^(-n) = 1/(a^n)
                                                                          1. a^n x a^m = a^(m+n)
                                                                            1. a^m / a^n = a^(m-n)
                                                                              1. (a^m)^n = a^(m x n)
                                                                                1. a^0 = 1
                                                                                2. a ^ (1/n) = n√a
                                                                                  1. a^ (m/n) = n√a^m
                                                                                  2. Curve sketching and transformations
                                                                                    1. any graph of the form y=x^n pass through (0,0) and (1,1)
                                                                                      1. y=f(x)
                                                                                        1. y=f(x) + a is a transformation a units upwards
                                                                                          1. y=f(x+a) is a transformation -a units to the right
                                                                                            1. y = f(ax) is a stretch sf 1/a parallel to x axis
                                                                                              1. y = af(x) is a stretch sf a parallel to y axis
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