Quadratic Functions and Inequalities

1. Quadratic Functions and Inequalities Graph and solve quadratic equations and inequalities Write quadratic equations and functions Analyze graphs of quadratic functions

2. Quadratic functions can be used to model real-world phenomena like the motion of a falling object. • Quadratic functions can be used to model the shape of architectural structures such as the supporting cables of a suspension bridge. • Can you think of examples?

3. 6.1 Graphing quadratic functions • Quadratic function – ax2 + bx + c. • Parabola – graph of a quadratic function • Y – intercept is c • Axis of symmetry and x-coordinate of the vertices - x = (-b/2a) • The y-coordinate is the maximum and minimum value.

4. 6.1 continued • Opens up and has a minimum value when a > o. • Opens down and has a maximum value when a < 0. • Minimum point and minimum value are not the same. Minimum value is the y-coordinate of the vertex. Minimum point is the ordered pair (x,y) of the vertex.

5. F(x) = x2 + 8x + 9 • Find the y-intercept • Find the axis of symmetry • Find the vertex • Find the maximum or minimum value • Does the graph open up or down? 9 -8/2 = -4 (-4,-7) -7, min. Opens up

6. 6.1 examples • Graph f(x) = x2 + 3x – 1 by making a table of values. Find the vertex point first. Then use 2 number less and 2 numbers greater.

7. 6.1 examples • f(x) = 2 – 4x + x2 a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Graph the function • 2, x = 2, 2 • b. (0,2),(1,-1),(2,-2),(3,-1),(4,2

8. 6.1 Classwork 1. f(x) = -x2 + 2x + 3 a. Determine whether the function has a maximum and minimum value. b. State the maximum and minimum value of the function. • Maximum value • 4

9. 6.1 Classwork see ex. 4 p. 289 2. A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $.50 increase in the price, he will sell about 10 fewer coffee mugs per month. a. How much should the owner charge for each mug in order to maximize the monthly income from their sales? b. What is the maximum monthly income the owner can expect to make from these items? $8 $1280

10. 6.1 Classwork Consider the equations –x2 + 8x – 16 and x2 – 8x + 16. • What is true about the equations? • What is true about the graphs of the equations? • They have the same solution • They open in opposite directions

11. 6.2 Solve quadratic equations by graphing • ax2 + bx + c = 0 • The solutions are called the roots or the zeros. The solutions are the x-intercepts. • Solutions of quadratic equations – 1 real solution or 2 real solutions or no real solution.

12. 6.2 examples -1,4 • Solve x2 -3x – 4 = 0 by graphing. • Solve x2 – 4x = -4 by graphing. • Find two numbers whose sum is 4 and whose product is 5 or show that no such numbers exist. • Solve x2 – 6x + 3 = 0 by graphing. Estimate. 2 No real solution Between 0 and 1, between 5 and 6

13. 6.3 solve quadratic equations by factoring • x2 = 6x x2 -6x = 0 x(x-6) = 0 x = 0 x – 6 = 0 x = 6 2. 2x2 + 7x = 15 2x2 + 7x – 15 = 0 (2x – 3)(x + 5) = 0 2x – 3 = 0 x + 5 = 0 x = 3/2 x = -5

14. 6.3 examples Solve by factoring 1. x2 = -4x 2. 3x2 = 5x + 2 3. x2 – 6x = -9 4. Write a quadratic equation with -2/3 and 6 as its roots. (see example 4 p. 303) {0,-4} {-1/3, 2} {3} 3x2 – 16x – 12 = 0

15. 6.4 Completing the square Solve using the square root property • x2 + 10x + 25 = 49 (x + 5)2 = 49 x + 5 = + x = -5 + 7 x = -12 x = 2

16. continued • x2 – 6x + 9 = 32 (x – 3)2 = 32 x – 3 = + x = 3 + 4 x = 8.7 x = -2.7

17. Complete the square • x2 + 8x – 20 = 0 x2 + 8x + ___ = 20 + ___ x2 + 8x + 16 = 20 + 16 (x + 4)2 = 36 x + 4 = +6 x = -4 + 6 x = -10, x = 2

18. continued 2. x2 + 4x + 11 = 0 x2 + 4x + _____ = -11 + ___ x2 + 4x + 4 = -11 + 4 (x + 2)2 = -7 x + 2 = No real solution

19. continued 3. 2x2 – 5x + 3 = 0 x2 – (5/2)x + (3/2) = 0 x2 – (5/2)x + ____ = -(3/2) + ___ x2 – (5/2)x + (25/16)= –(3/2)+ (25/16) (x – (5/4))2 = (1/16) x – (5/4) = +(1/4) x =+(1/4) + (5/4) x = 3/2 x = 1

20. 6.4 Examples Solve {-15,1} • x2 + 14x + 49 = 64 • x2 – 10x + 25 = 12 • x2 + 4x – 12 = 0 • x2 + 2x + 3 = 0 • 3x2 – 2x + 1 = 0 {5 + 2 } {-6,2} {no real solution} {-1/3,1}

21. 6.5 Quadratic Formula and discriminant • Quadratic formula • Discriminant b2 -4ac • b2 – 4ac > 0 2 real roots • b2 -4ac = 0 1 real root • b2 – 4ac < 0 no real roots

22. 6.5 Examples • Solve x2 – 8x = 33 using the quadratic formula • Solve x2 - 34x + 289 = 0 • Solve x2 – 6x + 2 = 0 • Solve x2 + 13x = 6x

23. 6.5 Examples continued • Find the value of the discriminant and describe the nature and types of roots. • x2 + 6x + 9 = 0 • x2 + 3x + 5 = 0 • x2 + 8x – 4 =0 • x2 – 11x + 10 = 0

24. 6.6 Analyzing Graphs • Vertex form y = a(x-h)2 + k • (h,k) vertex • x = h axis of symmetry • Translations h units to left if h is + h units to right if h is – k units up if k is + k units down if k is -

25. If a > 0, then the graph opens up • If a < 0, then the graph opens down • If abs(a) > 1, then the graph is narrower than y = x2 • If abs(a)< 1, then the graph is wider than y = x2

26. How to use vertex form to graph • Plot the vertex • Draw the axis of symmetry • Find and plot 2 points on one side of the axis of symmetry • Use symmetry to complete the graph

27. 6.6 Examples • Write y = x2 + 2x + 4 in vertex form. Analyze the function. • Write y = -2x2 – 4x + 2 in vertex form. Analyze and graph the function. • Write an equation for the parabola whose vertex is at (1,2) and passes through (3,4)

28. 6.7 Graphing and solving quadratic inequalities • Graph the inequality and decide if the parabola should be solid or dashed. • Test a point inside the parabola to see if it is a solution. • If it is a solution, shade inside. • If it is not a solution, shade outside.

29. 6.7 Examples • Graph y > x2 – 3x + 2 • Solve x2 – 4x + 3 > 0 by graphing. • Solve 0 < -2x2 – 6x + 1 by graphing. • Solve x2 + x < 2 algebraically.

30. Chapter 6 Study Guide • Solve by factoring • Solve by completing the square • Solve using quadratic formula • Solve using graphing calculator • Set up word problems using quadratic equations and solve • Solve using a method of your choice • Give the discriminant and describe the nature of the roots • Write a quadratic equation with the given roots.