Chapter 2 Summary: Quadratic functions and 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. 

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

x=(-b±√(b^2-4ac))/2a

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

• 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 

• 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)