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Quadratic functions

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

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Quadratic functions

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  1. Quadratic functions A. Quadratic functionsB. Quadratic equationsC. 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 = 122 + (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 Equation: Graph: PARABOLA Table:

  8. B. Quadratic equations

  9. B. Quadratic equations Example Group excursion Revenue equal to 1872? •  2x²+ 40x + 1722 = 1872 • 2x²+ 40x + 1722  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 Solving a quadratic equation - strategy 1: based on factoring • 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 (Section 0.8 – example 6 p36) • Solve 4x² - 17x + 15 = 0 • Solve 2 + 6 y + 9y² = 0 • Solve z² + z + 1 = 0 • Solve (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 p142-144)

  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), • Exercise 3, 7, 5 • rest of exercise 2 • Exercise 4, 6, 8 and 9

  24. A. Quadratic functions Supplementary exercises Exercise 7

  25. C. Quadratic inequalities

  26. Definition C. Quadratic inequalities 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 conclusion: x-2 or x7 interval notation: ]-,-2][7,[ Solve now inequality

  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

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