1. Section 2.2 Quadratic Functions

2. Thursday Bellwork 4 What does a quadratic function look like? 4 Do you remember the standard form? 4 How could we use your calculators to make finding the max and min of a parabola?

3. Graphs of Quadratic Functions

4. Vocabulary 4 Quadratic Function: a function that can be written in the form f(x) = ax 2 + bx + c, a ≠ 0. 4 Standard form: f(x) = a(x - h) 2 + k 4 Parabola: graph of quadratic function; U- shaped curve; symmetric 4 Symmetry: when folded, two sides match exactly

5. f(x) = a(x - h) 2 + k 4 Vertex: the point (h,k); highest or lowest point of parabola; on axis of symmetry 4 a: describes steepness and direction of parabola 4 Axis of symmetry: x = h; fold line that divides parabola into two matching halves 4 Minimum: if a > 0, parabola opens up; vertex is minimum point or lowest point of parabola; k is the minimum value 4 Maximum: If a < 0, parabola opens down; vertex the maximum point or lowest point of parabola; k is the maximum value

6. Minimum (or maximum) function value for a quadratic occurs at the vertex. 4 Identify the vertex of each graph. Find the minimum or maximum value. 4 A.B. y x 10 -10 y x 10 -10

7. Graphs of Quadratic Functions Parabolas Minimum Vertex Axis of symmetry Maximum

8. Review Solving Quadratic Equations Example: 3x 2 – 2x = 21 Get in standard form. 3x 2 – 2x – 21 = 0 Factor. (3x + 7)(x – 3) = 0 Set each factor equal to zero. 3x+7=0 or x–3 = 0 Solve.x = -7/3 or x = 3 Check.

9. If no “B” term, can use square root property to solve for zeros. For example, if x 2 = 16, then x = ±4. 4 Solve f(x) = 1 – (x – 3) 2.

10. When factoring or square root property won’t work, you can always use the QUADRATIC FORMULA Get in standard form: ax 2 + bx + c = 0 Substitute for a, b, and c in formula. Solve. Check. This works every time!!!

11. Try these. Solve using the Quadratic formula. 1) 3x 2 - 2x - 1 = 0 1) x 2 + 2x = -3

12. Try these. Solve using the Quadratic formula. 3) 3n 2 – 5n - 8 = 0 4) 10x 2 – 9x + 6 = 0

13. Try these. Find the x- intercept & y-intercept: 1) f(x) = (x+3) 2 2) f(x) = (x-4) 2 - 3

14. Try these. Find the x- intercept & y-intercept: 3) f(x) = x 2 + 4x + 7 4) f(x) =x 2 - 6x + 2

15. Graphing Quadratic Functions in Standard Form

16. Graphing Quadratic Functions f(x) = a(x - h) 2 + k 1. Determine whether the parabola opens up (a > 0) or down (a < 0). 2. Find the vertex, (h,k). 3. Find x-intercepts by solving f(x) = 0. 4. Find the y-intercept by computing f(0). 5. Plot the intercepts, vertex, and additional points. Connect with a smooth curve. Note: Use the axis of symmetry (x = h) to plot additional points.

17. Seeing the Transformations

18. Using Standard Form a<0 so parabola has a minimum, opens down

19. Example Graph the quadratic function f(x) = - (x+2) 2 + 4.

20. Example Graph the quadratic function f(x) = (x-3) 2 - 4

21. Homework problems: 4 Finish working all the odd problems on worksheet!

22. Friday Continuing Quadratic Functions!

23. Bellwork 1) Graph the quadratic function f(x) = - (x+1) 2 + 5. 2) Solve using quadratic formula -3x 2 - 2x = -1 THIS WILL BE COLLECTED ALONG WITH HW FROM LAST NIGHT!

24. QUICK HW REVIEW

25. Graphing Quadratic Functions in the Form f(x)=ax 2 =bx+c

26. f(x) = ax 2 + bx + c 4 If equation is not in standard form, you may have to complete the square to determine the point (h,k).

27. Solving Quadratic Equations by Completing the Square

28. Perfect Square Trinomials l Examples l x 2 + 6x + 9 l x 2 - 10x + 25 l x 2 + 12x + 36

29. Creating a Perfect Square Trinomial l In the following perfect square trinomial, the constant term is missing. X 2 + 14x + ____ l Find the constant term by squaring half the coefficient of the linear term. l (14/2) 2 X 2 + 14x + 49

30. THE FORMULA

31. Perfect Square Trinomials l Create perfect square trinomials. l x 2 + 20x + ___ l x 2 - 4x + ___ l x 2 + 5x + ___ 100 4 25/4

32. Solving Quadratic Equations by Completing the Square Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation

33. Solving Quadratic Equations by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

34. Solving Quadratic Equations by Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

35. Solving Quadratic Equations by Completing the Square Step 4: Take the square root of each side

36. Solving Quadratic Equations by Completing the Square Step 5: Set up the two possibilities and solve

37. Graph of

38. Finding the vertex when in ax 2 + bx + c form. Vertex =

39. Using the form f(x)=ax 2 +bx+c a>0 so parabola has a minimum, opens up

40. Example Find the vertex of the function f(x)=-x 2 -3x+7

41. Re-Cap Quadratic Equations & Functions ax 2 + bx + c = 0 or f(x) = ax 2 + bx + c, 4 “x” is squared. 4 Graph would be a curve. 4 Solutions are at the x-intercepts. The “zeros” are the solutions or roots of the equation. 4 To solve: factor & use zero product property, take square root of both sides (if no B term), or use other methods.

42. (a) (b) (c) (d)

44. Example Graph the function f(x)= - x 2 - 3x + 7. Use the graph to identify the domain and range.

45. Minimum and Maximum Values of Quadratic Functions

47. Example For the function f(x)= - 3x 2 + 2x - 5 Without graphing determine whether it has a minimum or maximum and find it. Identify the function’s domain and range.

48. Graphing Calculator – Finding the Minimum or Maximum Input the equation into Y= Go to 2 nd Trace to get Calculate. Choose #4 for Maximum or #3 for Minimum. Move your cursor to the left (left bound) of the relative minimum or maximum point that you want to know the vertex for. Press Enter. Then move your cursor to the other side of the vertex – the right side of the vertex when it asks for the right bound. Press Enter. When it asks to guess, you can or simply press Enter. The next screen will show you the coordinates of the maximum or minimum.

49. Applications of Quadratic Functions

50. Quadratic Regression on the Graphing Calculator MKwh 18.79 28.16 36.25 45.39 56.78 68.61 78.78 810.96 More on the next slide.

51. Quadratic Regression on the Graphing Calculator To see the scatter plot of these data points press 2 nd Y= to get STAT PLOT. Press ENTER on #1. If Plot 2-3 are ON, then change those to OFF. You can turn all plots off by pressing #4. Then return to this screen and press #1 to turn this plot on. By pressing GRAPH you will get the graph that you see at left. This is the Plot1 Screen. Press ENTER on the word On. Cursor down and choose the style of graph that you want. The first is a scatterplot. The XList should be L1, and YList L2. Choose one of the marks for your graph. For L1 or L2 press 2 nd 1 or 2 nd 2.

53. Example You have 64 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

54. Graphing Calculator Problem continued on the next slide

55. Graphing Calculator- continued

56. (a) (b) (c) (d)

57. Additional Examples:

58. See Example 4, page 279 Check Point 4, page 280: f(x) = 4x 2 - 16x + 1000 a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function’s domain and range.

59. Example 1- Application Problem A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball, in meters, t seconds after the hit is given by the quadratic function How long does it take for the baseball to reach its maximum height? What is the maximum height obtained by the baseball?

60. Solution Vertex:

61. A farmer has 600 feet of fencing to enclose a rectangular field. If the field is next to a river, so no fencing is needed on one side, find the dimensions of the field that will maximize the area. Find the maximum area. Example 2 - Application

62. Solution x x 600 – 2x Width (x) = Length(600-2x) = Max area = Area = (600  2x)(x) A(x) =  2x 2 + 600x Find Vertex – to find max. = 150 y =  2(150) 2 + 600(150) = 45000 150 300 45000

63. Friday