1. Quadratic FunctionsMenu IntroductionGraphing QuadraticsSolve by FactoringSolve by Square RootComplex NumbersCompleting the SquareQuadratic FormulaQuadratic Formula (graphtheory) (graphtheory) Quadratic InequalitiesModelingAppendix

2. Quadratic Functions Introduction MENU APPENDIX A QUADRATIC EQUATION is any equation of degree 2 That just means that there is an in the equation and no bigger exponents QUADRATIC EQUATIONS : STANDARD FORM FACTORED FORM INTERCEPT FORM VERTEX FORM or

3. Quadratic Functions Introduction MENU APPENDIX This equation is in standard form. Rewrite it in intercept form. This equation is in intercept form. Rewrite it in standard form. This equation is in vertex form. Rewrite it in standard form. Then it is easy to change to FACTORED FORM Then it is easy to change to FACTORED FORM

4. Quadratic Functions Graphing Quadratics MENU APPENDIX The graph of a quadratic equation looks like this: This U-shaped graph is called a PARABOLA The VERTEX is the high or low point on the graph It can cross the X-Axis 0, 1, or 2 times We call The X-Intercepts: Zeros, Roots, Solutions The Axis of Symmetry is the “mirror” that passes through the vertex It can open UP or DOWN

5. Quadratic Functions Graphing Quadratics MENU APPENDIX If “a” is positive it opens up If “a” is negative it opens down

6. Quadratic Functions Graphing Quadratics MENU APPENDIX Basic graphing method EXAMPLE #1 Graph the Quadratic Function using a 5- point table 1.Find the x- coordinate of the vertex. 2.Plug the x-coordinate into the equation to find the y-coordinate of the vertex 3.Count up and down by 1’s to fill in the left side of the table 4. Plug the numbers in to find the right side of the table.

7. Quadratic Functions Graphing Quadratics MENU APPENDIX Graph the Quadratic Function. Label the VERTEX and AXIS of SYMMETRY: Sketching a graph EXAMPLE #2 x = 3

8. Quadratic Functions Graphing Quadratics MENU APPENDIX Graph the Quadratic Function. Label the VERTEX and AXIS of SYMMETRY: Sketching a graph EXAMPLE #3 x = -4

9. Quadratic Functions Graphing Quadratics MENU APPENDIX Graph the Quadratic Function. Label the VERTEX, AXIS of SYMMETRY, and X-INTERCEPTS Sketching a graph EXAMPLE #4 x = 0

10. Quadratic Functions Graphing Quadratics MENU APPENDIX The height of a ball thrown straight up in the air, by an 8 foot tall person, at 30 meters per second is given by the following equation: EXAMPLE #5 What is the maximum height the ball reaches?

11. Quadratic Functions Factoring MENU APPENDIX MONOMIAL: 1 Term, a combination of numbers and variables being multiplied BINOMIAL: 2 Terms, two binomials being added together that are NOT “like terms” TRINOMIAL: 3 Terms, three binomials being added together that are NOT “like terms” Any expression with more than 3 “unlike terms” is called a POLYNOMIAL.

12. Quadratic Functions Factoring MENU APPENDIX POLYNOMIAL STANDARD FORM (for an expression) Like terms are combined Highest exponent first, then next highest, etc If there is more than 1 variable, write them alphabetically. Bad: Fixed:

13. Quadratic Functions Factoring MENU APPENDIX POLYNOMIAL STANDARD FORM (for an equation) Like terms are combined Highest exponent first, then next highest, etc If there is more than 1 variable, write them alphabetically. All the same rules, except, the equation must be equal to zero. (all terms on 1 side)

14. Quadratic Functions Factoring MENU APPENDIX Always do this first if possible

15. Quadratic Functions Factoring MENU APPENDIX

16. Quadratic Functions Factoring MENU APPENDIX

17. Quadratic Functions Factoring MENU APPENDIX

18. Quadratic Functions Factoring MENU APPENDIX

19. Quadratic Functions Square Roots MENU APPENDIX VOCABULARY: ROOT RADICAL RADICAND RATIONALIZE THE DENOMINATOR Short for “square root” Another name for square root The number or expression you are trying to take the square root of. Getting rid of a square root on the bottom of a fraction.

20. Quadratic Functions Square Roots MENU APPENDIX Click for Algebra I Square Root presentation

21. Quadratic Functions Square Roots MENU APPENDIX

22. Quadratic Functions Square Roots MENU APPENDIX

23. Quadratic Functions Square Roots MENU APPENDIX

24. Quadratic Functions Square Roots MENU APPENDIX

25. Quadratic Functions Complex Numbers MENU APPENDIX Getting ready for imaginary numbers

26. Quadratic Functions Complex Numbers MENU APPENDIX Taking the square root of a negative number No real number times itself will give you a negative So to take the square root of a negative, you have to factor -1 out Root -1 shows up any time we try to take the square root of a negative It’s not a real number, so we call it the IMAGINARY NUMBER: i

27. Quadratic Functions Complex Numbers MENU APPENDIX Vocab: Complex Number A Real and complex number added/subtracted together. The square root of -1 Imaginary Number Any number that ends with i. (i and it’s coefficient) Also called a “pure imaginary” number. Standard form for a Complex Number a+bi

28. Quadratic Functions Complex Numbers MENU APPENDIX Addition and Subtraction with Complex numbers The rules for adding, subtracting and multiplying and dividing complex numbers are the same for any variable

29. Quadratic Functions Complex Numbers MENU APPENDIX Simplifying an exponent on i Simplify each of the following:

30. Quadratic Functions Complex Numbers MENU APPENDIX Simplifying an exponent on i

31. Quadratic Functions Complex Numbers MENU APPENDIX Simplifying an exponent on i

32. Quadratic Functions Complex Numbers MENU APPENDIX Multiplication with Complex Numbers Just FOIL it out But i 2 is -1

33. Quadratic Functions Complex Numbers MENU APPENDIX 1. 2. 3. 4.

34. Quadratic Functions Complex Numbers MENU APPENDIX The Imaginary component disappeared! This happens anytime you multiply complex numbers like this ( a + bi ) ( a – bi ) They make the imaginary number disappear. So anytime you multiply two complex conjugates the answer will be a REAL NUMBER 4. These are called COMPLEX CONJUGATES

35. Quadratic Functions Complex Numbers MENU APPENDIX It is common and useful to graph them in what is called the COMPLEX PLANE We won’t go into why here. This is much easier than it sounds REAL IMAGINARY Graph the real number along the horizontal (x) and the imaginary number along the vertical (y)

36. Quadratic Functions Complex Numbers MENU APPENDIX GRAPH: R III R R

37. Quadratic Functions Complex Numbers MENU APPENDIX GRAPH: R III R R Every complex number has an ABSOLUTE VALUE. The absolute value is the length of the hypotenuse

38. Quadratic Functions Complex Numbers MENU APPENDIX STANDARD FORM: Complex numbers should always look like this: AND, since i is a square root, you cannot leave it on the bottom of a fraction

39. Quadratic Functions Complex Numbers MENU APPENDIX Put these into standard form #1 #2

40. Quadratic Functions Complex Numbers MENU APPENDIX Put these into standard form #3 #4 #5

41. Quadratic Functions Complex Numbers MENU APPENDIX Put these into standard form #6 #7

42. Quadratic Functions Complete the Square MENU APPENDIX Click for Complete the Square Presentation

43. Quadratic Functions Quadratic Formula MENU APPENDIX The Quadratic Formula Because the quadratic formula is simply a formula for completing the square… We can use it to sole ANY quadratic equation.

44. Quadratic Functions Quadratic Formula MENU APPENDIX The Quadratic Formula Even if the solutions are COMPLEX

45. Quadratic Functions Quadratic Formula MENU APPENDIX The graph tells us the types of solutions: #1 #2 #3

46. Quadratic Functions Quadratic Formula MENU APPENDIX Give the number and type of solutions based on the graph shown: #4 #5 #6 #7 #8 #9 1 REAL solution 2 COMPLEX solutions 2 REAL solutions 2 COMPLEX solutions 2 REAL solutions

47. Quadratic Functions Quadratic Formula MENU APPENDIX Determine the number and type of solutions without graphing: We can stop here, because we can already tell the solutions are going to be… COMPLEX 1 REAL 2 REAL

48. Quadratic Functions Quadratic Formula MENU APPENDIX The DISCRIMINANT If the discriminant is… POSITIVE… ZERO… NEGATIVE… There are 2 REAL solutions There is 1 REAL solution There are 2 COMPLEX solutions

49. Quadratic Functions Quadratic Formula MENU APPENDIX Describe the discriminant based on the graph: D: negative D: zero D: positive D: negative D: zero

52. Quadratic Functions Quadratic Inequalities MENU APPENDIX

53. Quadratic Functions Modeling MENU APPENDIX

54. Quadratic FunctionsAppendix MENU OPENERS ASSIGNMENTS EXTRA PROBLEMS

55. p. 253 Graphing Quadratic Equations #21 Graph, label the vertex and axis of symmetry. Vertex: X = 3

56. p. 253 Graphing Quadratic Equations #27 Graph, label the vertex and axis of symmetry. Vertex: X = 2

57. p. 253 Graphing Quadratic Equations #35 Graph, label the vertex and axis of symmetry AND X-INTERCEPTS Vertex: X = - 2. 5

58. p. 253 Graphing Quadratic Equations #39 Write the quadratic function in standard form. F O i L