Chapter 2 Summary: Quadratic functions and equations

A summary of the main points covered in the IB syllabus for standard level maths on quadratics

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Solving quadratic equations

  • If xy=0, then x=0 or y=0. 
    • This property is sometimes called the zero product property
  • The property can be expanded to:
    • If (x-a)(x-b)=0, then x-a=0, or x-b=0
  • To solve an equation by completing the square, take half the coefficient of x, square it, and add the result to both sides of the equation. This step creates a perfect square trinomial on the left side of the equation. 
  • In order to complete the square, the coefficient of the x^2 term must be 1. If the x^2 term has a coefficient other than 1, you can factor out the coefficient or divide through by the coefficient. 
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The quadratic formula

  • For any equation in the form ax^2+bx+c, 


This equation is given in the IB formula booklet, but it can be useful to memorize. 

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Roots of quadratic equations

  • For a quadratic equation (-b±√(b^2-4ac))/2a=0
    • If b^2 - 4ac>0, the equation will have two different real roots 
    • If b^2 - 4ac =0, the equation will have two equal real roots
    • If b^2 - 4ac <0, the equation will have no real roots 
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Graphs of quadratic equations

  • For quadratic functions in the standard form y=ax^2+bx+c=0, the graph will cross the y axis at (0,c)
  • The equation for the axis of symmetry is x=-b/2a
  • When the basic function y=x^2 undergoes a transformation, the resulting functions can be written as y=a(x-h)^2 +k 
  • For quadratic functions in the form y=a(x-h)^2 +k, the graph will have its vertex at (h,k)
  • For quadratic function in the form y=a(x-p)(x-q), the graph crosses the x axis at (p,0) and at (q,0)
  • For quadratic function in the form y=a(x-p)(x-q), the axis of symmetry will have the equation x=(p+q)/2
  • When the equation is in the form f(x)=a(x-h)^2+k, also known as the turning point form, the vertex will be (h,k). 
  • When the equation is written in factorized form f(x)=a(x-p)(x-q), the graph will cross the x axis at (p,0) and at (q,0)
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