Pure

• A proof is a logical and structured argument to show that a mathematical statement (or conjecture) is always true
• A statement that has been proven is called a theorem
• A statement that has yet to be proven is called a conjecture
• A mathematical proof needs to show that something is true in every case
• Steps to prove a mathematical statement

  1. State any information or any assumptions that you are using

  2. Show every step of your proof clearly

  3. Follow every step logically from the previous steps

  4. Cover all the possible cases

  5. Write a statement of proof at the end of your working
• Proof by deduction


  Starting from known facts or definitions, then using logical steps to reach the desired conclusion
• To prove an identity, you should: Start with the expression on one side of the identity, Manipulate that expression algebraically until it matches the other side, Show every step of your algebraic working
• The symbol ≡ means 'is always equals to'. It shows that two expressions are mathematically identical
• Proof by exhaustion: Break the statement into smaller cases and prove each case separately
• Counter-example


  One example that does not work for the given statement. To disprove a statement one counter example is enough
• Quadratic equation


  Equation in the form ax^2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0
• Solving quadratic equations


  1. Rewrite equation in the form ax^2 + bx + c = 0

  2. Factorise the left-hand side and set the factors equal to 0

  3. Find the values of x for each factor
• Quadratic equations can only have one, two or no real solutions
• Quadratic formula


  x = (-b ± √(b^2 - 4ac)) / (2a)
• Completing the square


  x^2 + bx + c = (x + b/2)^2 - (b/2)^2 + c
• Quadratic graph


  The plot of y = f(x) for a quadratic function f(x) = ax^2 + bx + c
• Quadratic graph


  • If a > 0, the graph is a U-shape
  • If a < 0, the graph is an inverted U-shape
• The roots of a quadratic function are where the graph intercepts the x-axis
• Discriminant


  The expression b^2 - 4ac, which indicates the number of real roots a quadratic function has
• If the discriminant b^2 - 4ac > 0, the function has two distinct real roots
• If the discriminant b^2 - 4ac = 0, the function has one repeated real root
• If the discriminant b^2 - 4ac < 0, the function has no real roots
• Translation of 𝑦 = 𝑓(𝑥) + 𝑎

  Can be represented by the vector (0

  𝑎)
• Translation of 𝑦 = 𝑓(𝑥 + 𝑎)

  Can be represented by the vector (−𝑎

  0 )
• Cubic functions


  Come in the form of 𝑓(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑
• Quartic functions


  Come in the form of 𝑓(𝑥) = ��𝑥4 + 𝑏𝑥3 + 𝑐𝑥2 + 𝑑𝑥 + 𝑒
• Sketching cubic graphs


  1. Determine the general shape of the cubic equation

  2. Locate the roots of the equation
• Reciprocal graphs


  Come in the form of 𝑓(𝑥) = 𝑎/𝑥 or 𝑓(𝑥) = 𝑎/𝑥2, where �� is any real number
• Reciprocal graphs


  • Have asymptotes at 𝑥 = 0 and 𝑦 = 0
• Stretching graphs


  Multiplying a constant outside, 𝑦 = 𝑎𝑓(𝑥), or inside, 𝑦 = 𝑓(𝑎𝑥), a function stretches the graph in the vertical direction or horizontal direction respectively
• 𝑦 = 𝑎��(𝑥)


  Stretches the graph in the vertical direction by a multiple of 𝑎
• 𝑦 = ��(𝑎𝑥)


  Stretches the graph in the horizontal direction by a multiple of 1/𝑎
• �� = −𝑓(𝑥)


  Reflection of 𝑦 = 𝑓(𝑥) in the 𝑥-axis
• 𝑦 = 𝑓(−𝑥)


  Reflection of 𝑦 = ��(𝑥) in the 𝑦-axis
• The graph of �� = −𝑓(𝑥) would be the reflection of 𝑦 = 𝑓(𝑥) in the 𝑥-axis
• The transformation of 𝑦 = 1/2 𝑓(𝑥) would stretch the graph by a multiple of 1/2
• Solving linear simultaneous equations

  Use method of elimination or substitution
• Quadratic simultaneous equations


  Equations where one is quadratic and one is linear, can have up to two solutions
• Simplifying algebraic fractions


  1. Cancel common factor

  2. Factorise the expression before cancelling common factor
• Graphing simultaneous equations


  Intersection point(s) represent the solution(s)