Core 1

Core 1

1. Chapter 1 ~ Algebra and Functions

   • Like terms
   • Multiplying brackets
   • Rules of Indices
   • Factorising
   • Surds
     • √(ab)=√(a)x√(b)
     • √(a/b)=√(a)/√(b)
     • Rationalising
       • Multiply top and bottom by denominator with opposite central sign

2. Chapter 2 ~ Quadratic Functions

   • y = f(x) = ax^2 + bx +c
   • Solve quadratic equations by factorisation
   • Completing the square
     • 1) Write formula > y=x^2 + bx + c
       • 2) Make a gap appear > y=x^2 + bx ________________ + c
         • 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
   • x = [-b±√((b^2)-4ac))]/2a
   • Discriminant
     • b^2>4ac and a>0
       • Two different roots
       • u shape crosses x-axis twice
     • b^2=4ac and a>0
       • Two equal roots
       • u shape sits on x-axis
     • b^2<4ac and a>0
       • No real roots
       • u shape that doesn't touch x-axis
     • b^2>4ac and a<0
       • Two real roots
       • n shape crosses x-axis twice
     • b^2=4ac and a<0
       • Two equal roots
       • n shape sits on x-axis
     • b^2<4ac and a<0
       • No real roots
       • n shape that doesn't touch the x-axis

3. Chapter 3 ~ Equations and Inequalities

   • Solving simultaneous linear equations
     • Elimination
     • Substitution
   • Solving simultaneous equations where one quadratic and one linear equation
     • Substitution
   • Inequalities
     • Solve similar to equations
     • Number lines
     • Sketches

4. Chapter 4 ~ Sketching Curves

   • Cubic Curves
     • y = ax^3 + bx^2 + cx + d
     • Use factors of equation to work out where the curve crosses the x-axis
       • x-axis intersections are when x = 0
         • eg (x-2) > intersection at (2,0). (x+5) > intersection at (-5,0)
     • y = x^3
       • Smooth curve through (0,0)
   • Reciprocals
     • y = k/x
       • When k>0, curves appear in quadrants where both values are either positive or negative
       • When k<0, curves appear in quadrants where one value is positive and the other is negative
     • The further away k is from 0, the further away the curves are from the axes
   • Transformations
     • f(x+a) > moves whole curve -a in the x-direction
     • f(x)+a > moves whole curve +a in y-direction
     • f(ax) > multiply x-coordinates by (1/a)
     • af(x) > multiply y-coordinates by a

5. Chapter 5 ~ Coordinate Geometry in the (x,y) Plane

   • y = mx + c
     • m is the gradient and c is the y-intercept
   • ax + by +c = 0
     • a, b, and c are all integers
   • Gradient between two points = (y2 - y1)/(x2 - x1)
     • Equation of a line using one point and the gradient > y - y1 = m(x - x1)
     • Equation of a line between two points > (y - y1)/(y2 - y1) = (x-x1)/(x2 - x1)
   • Two lines
     • Perpendicular
       • Gradient = -1/m
       • The product of two perpendicular lines is -1
     • Parallel
       • Same gradient

6. Chapter 6 ~ Sequences and Series

   • General term > nth term
     • a + (n-1)d
   • Un = 4n + 1
   • C1 only has arithmetic sequences
     • a > first term
     • d > common difference
   • Sum of an arithmetic sequence
     • Sn = (n/2)[2a + (n - 1)d]
     • sn = (n/2)(a + L)
       • L is the last term
     • ∑_(r=1)^10(5+2r) =7+9+...+25

7. Chapter 7 ~ Differentiation

   • Used to work out the gradient of a tangent
   • f(x) = x^n
     • f'(x) = nx^(n-1)
     • To get f'(x), multiply power by number in front of x, then reduce power by 1 for each part separately
   • (d^2y)/(dx^2) = f''(x)

8. Chapter 8 ~ Intergration

   • If dy/dx = x^n, the y = (1/(n+1))(x^(n+1)) + c
   • ∫x^n dx= x^(n+1)/(n+1)+c
   • Calculate c when given any point that the function of the curve passes through
   • Reverse of differentiation