IB Mathematics (HL)/Functions

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Topic 2: Core - Functions and EquationsEdit

The Axis of Symmetry for the Graph of a Quadratic FunctionEdit

 

The axis of symmetry is  

Ex.  

The axis of symmetry of the graph is  

Solving QuadraticsEdit

Quadratic Equations are in the form   or in the form  . To be solved the equations either have to be factored or be solved using the quadratic formula :  

Ex.   Since this cannot be factored, it is possible to use the quadratic formula  

DiscriminantEdit

The discriminant of the equation is important in determining whether the equation has 2, 1, 0 roots The equation of the discriminant:  

  : The equation has 2 real roots

  : The equation has 1 real root

  : The equation has 0 real roots

If the middle number is even in   then the discriminant can be calculated as  . The properties of this modified equation remain the same

Higher level FunctionsEdit

These functions have a degree of two or higher and as a result have more than 2 roots. An example of a higher polynomial function is y = x3 − 2x. This is a cubic equation, with three roots. To find these roots just factor the equation. In this case, it becomes, x(x2−2). From here you can factor using the difference of squares (a2−b2). Thus the equation then becomes, y=x(x+√2)(x−√2). The roots of the equation then become 0,±√2.