1. Quadratic functions
A. Quadratic functions B. Quadratic equations C. Quadratic inequalities

2. Quadratic functions

3. A. Quadratic functions Example Remember exercise 4 (linear functions):
For a local pizza parlor the weekly demand function
is given by p=26-q/40.
Express the revenue as a function of the demand q.
Solution:
revenue= price x quantity = 26q –q²/40

4. A. Quadratic functions Example
Group excursion
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 = (20 + 6)  (80  2  6) = 1890

5. A. 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 y of the travel agency when there are x supplementary participants?
QUADRATIC FUNCTION!

6. A. Quadratic functions Definition
A function f is a quadratic function if and
only if f(x) can be written in the form
f(x) = y=ax² + bx + c
where a, b and c are constants and a  0.
(Section 3.3 p141)

7. A. Quadratic functions Example Graph: PARABOLA Table: Equation: x y
1722 1
1760 2
1794 …

8. B. Quadratic equations

9. B. Quadratic equations Example Group excursion Revenue equal to 1872?
 2x² + 40x = 1872
2x² + 40x  1872 = 0
2x² + 40x  150 = 0
We have to solve the equation  2x² + 40x  150 =0
Quadratic equation

10. B. Quadratic equations Definition
A quadratic equation is an equation that can
be written in the form f(x) = y=ax² + bx + c
where a, b and c are constants and a  0.
(Section 0.8 p36)

11. B. Quadratic equations Exercises Solve x²+x-12=0 Solve (3x-4)(x+1)=-2
Solving a quadratic equation - strategy 1:
based on factoring
Exercises
Solve x²+x-12=0
Solve (3x-4)(x+1)=-2
Solve 4x-4x³=0
Solve
Solve x²=3
(Section 0.8 – example 1 p36)
(Section 0.8 – example 2 p37)
(Section 0.8 – example 3 p37)
(Section 0.8 – example 4 p37)
(Section 0.8 – example 5 p38)

12. B. Quadratic equations Solution Discriminant: d = b²  4ac
Solving a quadratic equation - strategy 2:
if discriminant d > 0: two solutions
if discriminant d = 0: one solution
if discriminant d < 0: no solutions

13. B. Quadratic equations Exercises Solve 4x² - 17x + 15 = 0
Solve y + 9y² = 0
Solve z² + z + 1 = 0
Solve
(Section 0.8 – example 6 p36)
(Section 0.8 – example 7 p37)
(Section 0.8 – example 8 p37)
(Section 0.8 – example 9 p37)
Supplementary exercises
Exercise 1

14. A. Quadratic functions

15. A. Quadratic functions Graph Quadratic functions: graph is a PARABOLA
What does the sign of a mean ?
If a>0, the parabola opens upward.
If a<0, the parabola opens downward
Example
Group excursion: y=-2x²+40x+1722
a<0
(Section 3.3 p )

16. A. Quadratic functions Graph Quadratic functions: graph is a PARABOLA
Graphical interpretation of y=ax²+bx+c=0 ?
Sign of the discriminant determines the number of intersections with the horizontal axis
Zero’s, also called x-intercepts, solutions of the quadratic equation y=ax²+bx+c=0 correspond to intersections with the horizontal x-axis
Example
Group excursion: y=-2x²+40x+1722
d=124²>0 x=41; (x=-21)

17. A. Quadratic functions Graph
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

18. A. Quadratic functions Graph Quadratic functions: graph is a PARABOLA
What is the Y-intercept ?
Example
Group excursion: y=-2x²+40x+1722
The y-intercept is c.

19. A. Quadratic functions Graph •
Each parabola is symmetric about a vertical line.
Which line ?
Both parabola’s at the right show a point labeled vertex, where the symmetry axis cuts the parabola.
If a>0, the vertex is the “lowest” point on the parabola. If a<0, the vertex refers to the “highest” point. •
x-coordinate of vertex equals -b/(2a)

20. A. Quadratic functions Example Group excursion: 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!

21. A. Quadratic functions Graph x-coordinate of vertex equals -b/(2a)
Example
Group excursion:

22. A. Quadratic functions Exercises
Graph the quadratic function y = -x² - 4x + 12.
Sign a? Sign d? Zeros?
Y-intercept? Vertex?
A man standing on a pitcher’s mound throws a ball
straight up with an initial velocity of 32 feet per
second. The height of the ball in feet t seconds
after it was thrown is described by the function
h(t)= - 16t²+32t+8, for t ≥ 0.
What is the maximum height?
When does the ball hit the ground?
(Section 3.3 – example 1 p143)
(Section 3.2 – Apply it 14 p144)

23. A. Quadratic functions Supplementary exercises Exercise 2 (f1 and f5),
rest of exercise 2
Exercise 4, 6, 8 and 9

24. A. Quadratic functions
Supplementary exercises
Exercise 7

25. C. Quadratic inequalities

26. C. Quadratic inequalities
Definition
A quadratic inequality is one that can be written in the form ax² + bx + c ‘unequal’ 0, where a, b and c are constants and a  0 and where ‘unequal’ stands for <, , > or .
Example
Solve the inequality
i.e. Find all x for which
standard form

27. C. Quadratic inequalities
Example
Study the equality first:
Next, determine type of graph
x=-2; x=7
Solve now inequality
conclusion: x-2 or x7
interval notation: ]-,-2][7,[

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

29. C. Quadratic inequalities
Supplementary exercises
Exercise 10 (a)
Exercises 11 (a), (c)
Exercises 10 (b), (c), (d)
Exercises 11 (b), (d)

30. Quadratic functions Summary
Quadratic equations : discriminant d, solutions
Quadratic functions :
Graph: Parabola, sign of a,
sign of d, zeros
vertex, symmetry axis, minimum/maximum
Quadratic inequalities : solutions
Extra: Handbook –
Problems 0.8: Ex 31, 37, 45, 55, 57, 79
Problems 3.3: Ex 11, 13, 23, 29, 37, 41