1. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 6.3 The quadratic function

2. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function The quadratic function takes the form f(x) = ax 2 + bx + c where a ≠ 0. The graph of a quadratic function has a characteristic shape. Whether the graph is or depends on the value of a as shown below. a > 0 a < 0

3. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function When the quadratic function is graphed, there are several key properties that need to be identified. These are as follows. axis of symmetry y-intercept x-intercepts vertex

4. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function At the y-intercept, the x-coordinate is 0. Therefore the value x = 0 can be substituted into the equation y = ax 2 + bx + c. This gives y = c. y-intercept

5. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function At the x-intercept, the y-coordinate is 0. Therefore the value y = 0 can be substituted into the equation y = ax 2 + bx + c. This gives ax 2 + bx + c = 0. x-intercepts

6. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function ax 2 + bx + c = 0 can sometimes be solved by factorizing to give a(x + p)(x + q) = 0. If it cannot be factorized, x can be found by using the quadratic formula

7. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function The axis of symmetry for the quadratic y = ax 2 + bx + c is found using the equation Because the graph is symmetrical, the axis of symmetry also passes through the point that is halfway between the two x-intercepts. axis of symmetry x-intercepts

8. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 The quadratic function The vertex lies on the axis of symmetry, therefore its x-coordinate is also found by the formula To find the y-coordinate of the vertex, substitute the x-value found above into the equation of the quadratic y = ax 2 + bx + c. vertex axis of symmetry