A Maths

• Quadratic functions are polynomial functions of the second degree, expressed as 𝒇 𝒙 = 𝒂 𝒙 𝟐 + 𝒃𝒙 + 𝒄
• 𝒂 and 𝒃 are coefficients of the 𝒙 𝟐 and 𝒙 terms respectively
• Maximum/minimum points are also known as the "Turning Point" or "Vertex" of a graph
• The line of symmetry of the graph can be found by adding the 𝒙-intercepts of the graph and dividing the total value by 2
• To find the maximum/minimum points of a quadratic function, completing the square is one method
• The shape of the quadratic graph is based on the signature of the quadratic coefficient 𝒂:

  • 𝒂 > 0 results in a "Happy-face curve"
  • 𝒂 < 0 results in a "Sad-face curve"
• Conditions for positivity/negativity:

  • No Real Roots: 𝑫 = 𝒃 𝟐 - 𝟒𝒂𝒄 < 0
  • The signature of 𝒂 changes based on whether it is positive or negative
• Conditions for a quadratic equation to have:

  • Two real roots
  • Two equal roots
  • No real roots
• Relating conditions for a given line to:

  • Intersect a given curve
  • Be a tangent to a given curve
  • Not intersect a given curve
• Solving quadratic inequalities and representing the solution on the number line
• A quadratic function equated to a number is a quadratic equation
• Factorisation:

  • Factorise the quadratic equation
  • Solving the equation gives x = α or x = β
• Quadratic Formula:

  • The roots of the equation ax^2 + bx + c = 0 can be obtained with:

  x = (-b ± √(b^2 - 4ac))
• Nature of Roots:

  • The expression (b^2 - 4ac) is known as the discriminant/determinant of the quadratic equation
  • The nature of the roots can be discriminated by examining the discriminant
  • Discriminant:
  • 2 real and distinct roots: b^2 - 4ac > 0
  • 2 equal roots: b^2 - 4ac = 0
  • No real roots: b^2 - 4ac < 0
  • Special case for "2 real roots": b^2 - 4ac ≥ 0
• Line intersecting Curves:

  • Solve the equations simultaneously by substituting the equation of the line into the equation of the curve to eliminate one of the variables
  • The number of intersection points can be identified by examining the discriminant
  • Intersection Discriminant:
  • Intersect at 2 distinct points: b^2 - 4ac > 0
  • Tangential: b^2 - 4ac = 0
  • Do not intersect: b^2 - 4ac < 0
• Quadratic Inequality Properties:

  • Let a, b, c, and d be any 4 real numbers
  • When multiplying/dividing by a negative number, remember to reverse the direction of the inequality
• Expressing Quadratic Inequalities:

  • Sketch out the graph to see where the critical regions related to the inequality are
  • Rewrite the inequality into the form
  • Factorise the equation to get into the form
  • Solve the equation to get the roots of the equation x = α or x = β
  • To figure out the critical regions in relation to the inequality, draw the number line and sketch the curve
  • Observe the direction of the inequality symbol