1. Quadratic functions

2. Quadratic functions: example Minimum 20 participants Price of the guide: 122 EUR For 20 participants: 80 EUR per person For every supplementary participant: for everybody (also the first 20) a price reduction of 2 EUR per supplementary participant Revenue of the travel agency when there are 6 supplementary participants? total revenue = 122 + (20 + 6)  (80  2  6) = 1890 Group excursion

3. Minimum 20 participants Price of the guide: 122 EUR For 20 participants: 80 EUR per person For every supplementary participant: for everybody (also the first 20) a price reduction of 2 EUR per supplementary participant Revenue y of the travel agency when there are x supplementary participants? y = 122 + (20 + x)(80  2x) =  2x² + 40x + 1722 QUADRATIC FUNCTION! Quadratic functions: example Group excursion

4. Equation: Graph: Table: xy 01722 11760 21794 …… Quadratic functions: 3 representations PARABOLA

5. Quadratic functions: equation A function f (“rule”) with an equation of the form f(x) = y=ax² + bx + c where a  0 is called a quadratic function

6. Quadratic equations  2x² + 40x + 1722 = 1872  2x² + 40x + 1722  1872 = 0  2x² + 40x  150 = 0 We have to solve the equation  2x² + 40x  150 = 0. Finding SOLUTIONS of an equation of the form ax² + bx + c = 0 QUADRATIC EQUATION Revenue equal to 1872?

7. Quadratic functions: definitions Function f (“MACHINE”!) with an equation of the form f(x) = ax² + bx + c where a  0. Or: function having an explicit equation of the form y = ax² + bx + c where a  0. Discriminant: d = b²  4ac

8. Quadratic equations Equations that can be written in the form ax² + bx + c = 0 where a  0. Solutions: if discriminant d > 0: two solutions if discriminant d = 0: one solution if discriminant d < 0: no solutions Group excursion:  2x² + 40x  150 = 0 Discriminant: d = b²  4ac

9. Quadratic functions: exercises exercise 1 (a), (c) and (e) supplementary exercises: rest of exercise 1

10. Quadratic functions: graph is a PARABOLA Quadratic function: graph Sign of the discriminant determines the number of intersections with the horizontal axis Graphical interpretation of y=ax²+bx+c=0 ? Zero’s, solutions of this quadratic equations correspond to intersections with the horizontal x-axis

11. sign of the discriminant determines the number of intersections with the horizontal axis sign of the coefficient of x 2 determines the orientation of the opening Quadratic function: graph

12. Maximum revenue? In this case you can find it e.g. using the table: So: 10 supplementary participants (30 participants in total) This can also be determined algebraically, based on a general study of quadratic functions! Quadratic function: maximum

13. x-coordinate of the vertex of the parabola: the vertex determines the minimum/maximum function value Group excursion: Quadratic function: maximum

14. Exercise 2 (f 1 and f 5 ) Exercise 3, 7, 5 supplementary exercises: rest of exercise 2 exercise 4, 6, 8 and 9 Figure Quadratic functions: exercises

15. Quadratic inequalities Solve the inequality i.e. Find all x for which standard form graph of LHS conclusion: x  -2 or x  7 interval notation: ]- ,-2]  [7,  [

16. Quadratic inequalities inequalities that can be reduced to the form... and determine the common points with the x-axis by solving the EQUATION

17. Exercise 10 (a) Exercises 11 (a), (c) Supplementary exercises: Exercises 10 (b), (c), (d) Exercises 11 (b), (d) Quadratic functions: exercises

18. 3 representations : table, equation, graph Quadratic equations, zero’s, discriminant d Graph: Parabola interpretation of d, a Maximum, vertex Quadratic inequalities Quadratic functions: Summary

19. STUDYING MATH is DOING A LOT OF EXERCISES YOURSELF, MAKING MISTAKES AND DOING THE EXERCISES AGAIN CORECTLY

20. Exercise 7 Back