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

2 A. Quadratic functions

3 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 ♦ Fixed cost: 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  6  2) = 1890

5 Example Group excursion ♦ Minimum 20 participants ♦ Fixed cost: 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? total revenue = y= (20 + x)  (80  x  2) = 2x x A. Quadratic functions

6 revenue function y = 2x x is a quadratic function Definition A function f is a quadratic function if and only if its equation 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 p. 141)

7 A. Quadratic functions Example Equation: Graph: xy …… Table: PARABOLA

8 B. Quadratic equations

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

10 B. Quadratic equations Definition A quadratic equation is an equation that can be written in the form ax² + bx + c = 0 where a, b and c are constants and a  0. (Section 0.8 p. 36)

11 B. Quadratic equations Solving a quadratic equation - strategy 1: based on factoring, … Exercises 1.Solve x²+x12=0 2.Solve (3x  4)(x+1)= 2 3.Solve 4x  4x³=0 4.Solve 5.Solve x²=3 (Section 0.8 – example 1 p. 36) (Section 0.8 – example 2 p. 37) (Section 0.8 – example 3 p. 37) (Section 0.8 – ex. 4 p. 37) (Section 0.8 – example 5 p. 38)

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

13 B. Quadratic equations Exercises 1.Solve 4x² - 17x + 15 = 0 2.Solve 2x² + 40x  150 = 0 3.Solve y + 9y² = 0 4.Solve z² + z + 1 = 0 5.Solve Supplementary exercises: exercise 1 (Section 0.8 – example 6 p. 36) (Section 0.8 – example 7 p. 37) (Section 0.8 – example 8 p. 37) (Section 0.8 – example 9 p. 37) (example slide 9)

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 a = 2 < 0 (Section 3.3 p )

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

17 A. Quadratic functions sign of the discriminant determines the number of intersections with horizontal axis sign of the coefficient of x 2 determines the orientation of the opening Graph

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

19 A. Quadratic functions 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) axis of symmetry is line x=-b/(2a) Graph x = -b/(2a)

20 A. Quadratic functions x-coordinate of vertex equals -b/(2a) Group excursion: Maximum revenue? Example vertex is “highest” point maximum revenue = y-coo of vertex = y(10) = 1922

21 A. Quadratic functions Group excursion: Maximum revenue? In this case you can find it e.g. using the table: So: 10 supplementary participants (30 participants in total) Example

22 A. Quadratic functions Exercises 1.Graph the quadratic function y = -x² - 4x Sign a? Sign d? Zeros? Y-intercept? Vertex? 2.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 initial height of the ball? What is the maximum height? When is the ball back at a height of 8 feet? (Section 3.3 – example 1 p. 143) (Section 3.2 – Apply it 14 p. 144)

23 A. Quadratic functions Supplementary exercises Exercise 2 (f 1 and f 5 ), Exercise 3, 7, 5 rest of exercise 2 Exercise 4, 6, 8 and 9

24 A. Quadratic functions Exercise 7 Supplementary exercises lineparabola

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 Example C. Quadratic inequalities 2. Study the equality: 4. Solve inequality: x = -2; x = 7 conclusion: x-2 or x7 interval notation: ]-,-2][7,[ 3. Determine type of graph: 1. Write in standard form:

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

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

30 Quadratic functions Summary Quadratic equations: discriminant d, solutions Quadratic functions: ♦ graph: parabola ♦ sign of a ♦ sign of d ♦ zeroes ♦ vertex ♦ symmetry axis ♦ minimum/maximum Quadratic inequalities: solutions

31 Quadratic functions Extra exercises: Handbook – Problems 0.8: Ex 31, 37, 45, 55, 57, 79 Problems 3.3: Ex 11, 13, 23, 29, 37, 41