Mathematics As&A levels Pure 1

• Completing the Square


  The equation written in the form of p(x + q)^2 + r is called the complete square form
• Vertex


  (-q, r)
• Express 3x^2 + 9x + 5 in the form of p(x + q)^2 + r

  • p = 3, q = -3/2, r = 4/7
• Quadratic Graph


  • x-intercept
  • y-intercept
  • Vertex (turning point)

  
• Discriminant


  Describes the roots for a quadratic equation ax^2 + bx + c = 0

  
• If discriminant b^2 - 4ac = 0, real and equal (repeated) roots

  
• If discriminant b^2 - 4ac < 0, no real roots
• If discriminant b^2 - 4ac > 0, real and distinct roots
• Finding inequality for distinct real roots

  Apply b^2 - 4ac > 0 for the equation to have two distinct real roots

  
• Quadratic Inequalities


  • Case 1: Assuming a > 0
  • Case 2: When no x term

  
• Factorize kx^2 + 4kx + 3k = 0

  • k < 0 and k > 0
• The blue region represents the section of the parabola where the value of the quadratic is < 0

  
• Solving Equations in Quadratic Form

  1. Let u = x^2 to convert to a quadratic equation in u

  2. Reject solutions that don't satisfy the original equation
• Function


  A relation that uniquely associates one set of values to another set
• Domain


  The set of values that are the inputs of the function
• Range


  The set of values that are the outputs of the function
• Inverse Function


  The function which maps the Range back into its Domain
• Mapping


  • One-to-many
  • Many-to-one
  • One-to-one
  • Many-to-many
• Notations


  f(x), g(x) or f: x ↦ 2x + 5
• Finding Range


  For quadratic functions, complete the square first to find the vertex and use it to find the range

  
• One-One Functions


  One x value substitutes to give one y value
• Finding Inverse


  1. Make sure it is a one-to-one function

  2. Write f(x) as y

  3. Make x the subject

  4. Swap every x with y
• Graph of inverse function


  Reflection of the graph of the function in the line y = x

  
• Translating a Function


  1. Shift along x-axis by a units to the right

  2. Shift along y-axis by b units upwards
• Stretching a Function


  1. Stretch sideways by factor a (a > 1 expands, 0 < a < 1 shrinks)

  2. Stretch upwards and downwards by factor a (a > 1 expands, 0 < a < 1 shrinks)
• Reflecting a Function


  1. Reflection in x-axis: y = -f(x)

  2. Reflection in y-axis: y = f(-x)
• Length of a Line Segment

  Distance between two points (x1, y1) and (x2, y2)
• Finding Midpoint


  Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
• Equation of a Straight Line

  y = mx + c, where m is the gradient and c is the y-intercept

  
• Finding Equation from a Point and Gradient

  y - y1 = m(x - x1)

  
• Gradient


  The slope of a line
• General point on a line


  (x, y)
• Equation of a line

  1. Given a point and gradient

  2. Substitute into y = mx + c
• Example equation of a line


  • y = 5x + 2
• Transforming a function


  1. Shift horizontally by adding/subtracting

  2. Shift vertically by multiplying/dividing
• Transformed function examples


  • y = 2f(x)
  • y = 2(5x + 2)
  • y = 10x + 4
  • y = x + 2/(3x + 2)
  • y = f(3x)
  • y = (3x) + 2/(3(3x) + 2)
  • y = 9x + 2
  • y = f(x)
  • y = -f(x)
  • y = f(-x)
  • y = 2x + 2/(3x + 2)
  • y = f(-x)
  • y = 2(-x) + 2/(3(-x) + 2)
  • y = -2x - 2/(3x - 2)
  • y = -f(x)
• General point on a line

  (x1, y1), (x2, y2)
• Example points


  • P(0, 4), Q(a, 1)
• Example distance calculation


  • PQ = √((a - 0)^2 + (1 - 4)^2) = √(a^2 + 9) = 5
• a = 4