C1

C1

1. Simultaneous Equations & Disguised Quadratics

   • Linear
     • Add/ subtract the two equations to eliminate one variable
     • Set one equal to y or x and substitute in
   • Quadratic
     • Substitution
   • Disguised Quadratics
     • Some equations- eg x^4-x^2-5=0 can be converted to quadratic equations to solve more easily
     • If there are three orders, ie x^4, x^2, x^0
     • let A = the middle order, therefore highest order = A^2

2. Quadratics

   • eg 4x^2+3x+7=0
   • Solving
     • If the equation factorises
       • Each bracket = 0
       • eg.(4x+3)(x-2)=0
         • x = -3/4 or x = 2
     • If does not factorise...
       • Quadratic formula
         • x= (-b ± √(b^2-4ac) ) / 2a
       • Complete the square
         • x^2 ± bx = (x±b/2)^2 - (b/2)^2
           • Rearrange to find one or two values of x
         • a(x+b)^2 + c
           • Vertex = (-b, c )
   • Inequalities
     • If multiplying or dividing by a negative number, REVERSE the sign
     • Quadratic
       • Set so that equation = 0
       • Factorise
       • If equation > 0 it is where the graph is above the x axis
       • If equation < 0 it is where the graph is below the x axis
   • Intersections of lines
     • Set equal to eachother to eliminate y
     • Remember to get the y values at the end by re-substituting the x values

3. Gradients, tangents and normals

   • To find a gradient, differentiate the equation and then substitute in the x value
   • The tangent to a curve has the same gradient as the point on the curve it touches
   • y+y-value= m (x + x-value)
   • Stationary points
     • when dy/dx = 0
     • solve dy/dx=0 to find stationary points
     • 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

4. Coordinate Geometry, Lines and Circles

   • Midpoints, gradients and distance between two points
     • Point A => (x,y) Point B => (w,z)
       • midpoint = ( (x+w)/2 , (y+z)/2 )
       • length of the line through AB = √{ (x+w)^2 + (y+z)^2 }
       • Gradient = (x-w)/(y-z)
   • equation of a line through (a,b) with gradient m is y-b = m(x-a)
   • Circles
     • Equation of a circle centre (a,b) radius r = (x-a)^2 + (x-b)^2 = r^2

5. Surds and indices

   • Surds
     • √m x √n = √mn
     • √m / √n = √(m/n)
     • To simplify k/√a multiply by √a / √a
   • Indices
     • a^(-n) = 1/(a^n)
     • a^n x a^m = a^(m+n)
     • a^m / a^n = a^(m-n)
     • (a^m)^n = a^(m x n)
     • a^0 = 1
   • a ^ (1/n) = n√a
   • a^ (m/n) = n√a^m

6. Curve sketching and transformations

   • any graph of the form y=x^n pass through (0,0) and (1,1)
   • y=f(x)
     • y=f(x) + a is a transformation a units upwards
     • y=f(x+a) is a transformation -a units to the right
     • y = f(ax) is a stretch sf 1/a parallel to x axis
     • y = af(x) is a stretch sf a parallel to y axis