1. Unit 4 Quadratics

2. Quadratic Functions
Any function that can be written in the form

3. Put in Standard Form and Find a, b, and c

4. Is it quadratic?

5. Quadratic Functions
Graph forms a parabola
concave up concave down or

6. Determine whether a parabola opens up or down

7. Up or Down? Max or Min?

8. Using a graphing calculator, find vertex, line of symmetry, max/min, and zeros and where the function increases and decreases..

9. Another

10. And another

11. Average Rate of Change of a Quadratic

12. Example

13. Finding Average Rate of Change

14. Average Rate of Change

15. To find the axis of symmetry
When

16. Find the vertex and los

17. Vertex (h,k) form of a Quadratic
Standard Form:

18. Parent Function

19. Transformations
You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form

20. Sketch the graph

21. Sketch the graph

22. Sketch the graph

23. Sketch the graph

24. Sketch the graph

25. Identifying Important Parts on Calculator
2nd calc—then select max or min

26. Completing the Square
Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve
Steps:
Move the constant
Factor out the # in front of x2
Take ½ of middle term and square it
Write in factored form for the perfect sq. trinomial
Add to both sides (multiply by # in front)
Move constant back to get in (h,k) form

27. Solve

28. Solve

29. Complete the Square

30. Complete the Square

31. Example

32. Example

33. Example

34. Solving Quadratics
You can solve by graphing, factoring, square root method, and quadratic formula
Solutions, roots, or zeros

35. Solving by Graphing Graph the parabola
Look for where is crosses the x-axis (where y=0)
May have 2 real, 1 real, or no real solutions
(Show on calculator)
Review finding the vertex

36. Solve the following by graphing

37. Solving Quadratics by Factoring
Factor the quadratic
Set each factor that contains a variable equal to zero and solve (zero product property)

38. More solving by factoring

39. You Try

40. Writing the Quadratic Eq.
Write the quadratic with the given roots of ½ and -5

41. Write the quadratic with
Roots of 2/3 and -2

42. More about solving
Graphing—not always best unless you have exact answers
Factoring—not every polynomial can be factored
Quadratic Formula—always works
Square Root method—may have to complete the square first

43. Solving using Quadratic formula
Must be in standard form
Identify a, b, and c

44. Examples

45. Examples

46. Examples

47. Examples

48. Discriminant
Used to identify the “type” of solutions you will have (without having to solve)

49. If the discriminant is…
A perfect square---2 rational solutions
A non-perfect square—2 irrational sol.
Zero—1 rational sol.
Negative—2 complex sol.

50. Identify the nature of the solution

51. Identify the nature of the solution(s)

52. Solving Quadratics using the Sq. Rt. method
Useful when you have x2 = constant or a perfect sq. trinomial ex. (x-3)2=constant
Get the x2 by itself
Take the square rt. of both sides
Don’t forget + or – in your answer!!!

53. Examples

54. Examples

55. Examples

56. Quadratic Inequalities
Graphing quadratic inequalities in 2 variables:
Steps:
Graph the related equation
Test a point not on the graph of the parabola
Shade region that contains the point if it makes the inequality true or shade the other region if it does not make the inequality true
Ex Ex.

57. Graphing Quadratic Inequalities

58. Solving Quadratic Inequalities
Solving Quadratic Inequalities in one variable: May be solved by graphing or algebraically.
To solve by graphing:
Steps:
Put the inequality in standard form
Find the zeros and sketch the graph of the related equation
identify the x values for which the graph lies below the x-axis if the inequality sign is < or
identify the x values for which the graph lies above the x-axis if the inequality sign is > or

59. Solve by graphing Solutions:_______________________

60. To solve algebraically:
Steps:
Solve the related equation
Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set
Test all regions of the number to determine other values to include in the solution set

61. Solve Algebraically

62. Solving Quadratic Inequalities

63. Word Problems