1. SOLVING QUADRATIC EQUATIONS A.4c: The student will solve multi-step linear and quadratic equations in two variables, including…solving quadratic equations algebraically and graphically

2. SOLVING QUADRATIC EQUATIONS A.7b-f: The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including … Domain and range Zeros of a function x and y-intercepts Finding the values of a function for elements in it’s domain Making connections between and among multiple representations of functions including verbal, numeric, graphical, and algebraic.

3. CHECKLIST: 1.Graphing Quadratics Numerically (1 Day) 2.Graphing (1 Day) 3.Finding Roots (6 Days) 1.Solving By Square Root (1 Day) 2.Zero Product Property and Factoring (2 Days) 3.Completing The Square (1 Day) 4.Quadratic Formula (2 Days) 4. Word Problems (1 Day) 5. Review (1 Day) 6. Test (1 Day)

4. GRAPHING QUADRATICS NUMERICALLY When we first started studying functions, we looked at them as a rule that transformed some input to it’s output. Now, we introduce another rule, called a quadratic function. In this video, we will manually find a series of inputs and outputs given a certain rule.

5. NUMERICALLY FINDING QUADRATIC FUNCTIONS A quadratic function is a function that is of 2 nd degree, meaning, that the variables in the expression is either 1 or 2. …where a, b are coefficients and c is a constant term.

6. NUMERICALLY FINDING QUADRATIC FUNCTIONS

7. xf(x) -2 0 1 2 19 9 3 1 3

8. NUMERICALLY FINDING QUADRATIC FUNCTIONS xf(x) -2 0 3 6 10 80 20 5 80 320

9. NUMERICALLY FINDING QUADRATIC FUNCTIONS xf(x) -8 -5 0 5 8 60 21 -4 21 60

10. NUMERICALLY FINDING QUADRATIC FUNCTIONS So far, what characteristics have we noticed about a quadratic function… 1. Each quadratic function has a “turning point.” 2. Each quadratic function is symmetrical. Vertex Point

11. TRY IT PROBLEM

12. THE PARABOLA In our last video, we graphed a quadratic equation the good ‘ol fashioned way: arbitrarily picking some x values, plugging them into our quadratic function, then using the resulting coordinate points to plot the graph. In this video, we will talk more about what a parabola (the graph of a quadratic function) is and it’s characteristics.

13. THE PARABOLA A linear function graphs a line A quadratic function graphs a parabola

14. THE PARABOLA Axis of Symmetry An imaginary vertical line that cuts a parabola in half and on either side of this line is a mirror image.

15. THE PARABOLA What is the axis of symmetry? x = 3

16. THE PARABOLA Vertex The “turning point” or the point where the parabola crosses it’s axis of symmetry. It’s either the maximum or minimum point on the parabola.

17. THE PARABOLA x = 3 The vertex point (in this case, a maximum) is located at the point (3, 12)

18. THE PARABOLA x = 3 In general, to find the vertex point…. 1. Find the Axis of Symmetry (3, ) 2. Plug in that x value back into the equation to get the y value. 12

19. THE PARABOLA X-intercepts The points where the function crosses the x- axis, also known as… Roots Zeros Solutions

20. THE PARABOLA How many solutions can a quadratic function have? 0 1 2 Doesn’t intersect the x-axis Only the vertex intersects the x-axis Intersects the x-axis twice

21. THE PARABOLA Y-intercept The points where the function crosses the y-axis. y-coordinate of the y- intercept point

22. THE PARABOLA (0, 3)

23. TRY IT PROBLEM 1.What is the equation for the axis of symmetry for this function? 2.What is the vertex point? 3.Is the vertex point a max or a min? How can you tell? 4.What is the y-coordinate?

24. SOLVING BY SQUARE ROOTS In this video, we will look at the first case of solving (or finding the zeros/roots/solutions of) a quadratic equation. In this case, there is no factoring required: the form of the quadratic that we are given only requires us to take the square root and solve!

25. SOLVING BY SQUARE ROOTS Let’s say we are given the function… …and we want to find the x-intercepts. Roots Zeros Solutions

26. SOLVING BY SQUARE ROOTS and Examples

27. TRY IT PROBLEMS Find the roots of the following quadratic equation. Leave your answer in simplest radical form (that is, no decimals).

28. SOLVING QUADRATICS: ROOTS AND ZERO PRODUCT PROPERTY Recall from the linear functions unit that we studied x and y intercepts: the points at which the line intersected the x and y axis, respectively. This too is a relevant concept in quadratics, except here we pay more attention to the x-intercepts or the root(s) of a quadratic equation. The “solution” to a quadratic can be found at this location, and in this video, we will look at the Zero Product Property and how it relates to the roots of a quadratic function.

29. THE ZERO PRODUCT PROPERTY What is the value of the variable x in all of this equations? No Algebra skills required here…the answers have to be 0. NOTICE THAT ALL OF THE MULTIPLES OF X ARE CONSTANTS, THAT IS, THEY DON’T DEPEND ON ANYTHING.

30. THE ZERO PRODUCT PROPERTY What is the value of the variable x in all of this equations? NOTICE THAT ALL OF THE MULTIPLES OF X ARE NOT CONSTANTS, THAT IS, THEY DEPEND ON THE VALUE OF THE VARIABLES CONTAINED IN THEM.

31. THE ZERO PRODUCT PROPERTY What is the value of the variable x in all of this equations?

32. THE ZERO PRODUCT PROPERTY THIS MEANS THAT r AND k ARE BOTH ZERO IN THIS CASE. BUT DO BOTH OF THEM HAVE TO BE ZERO FOR THIS CONDITION TO BE SATISFIED?

33. THE ZERO PRODUCT PROPERTY Given the equation… The following options are possible: 1. a is equal to 0 2. b is equal to 0 3. Both a and b are equal to 0

34. QUADRATICS AND THE ZERO PRODUCT PROPERTY Want to know the x-intercept point? Set y to zero! …now what?

35. QUADRATICS AND THE ZERO PRODUCT PROPERTY Let’s try factoring… A product of two quantities

36. QUADRATICS AND THE ZERO PRODUCT PROPERTY Either… (x – 2) is 0 (x – 3) is 0 Both are 0 THIS LOGIC GIVES US TWO CONCLUSIONS: X IS BOTH 2 AND 3. HOW CAN THIS BE TRUE?

37. QUADRATICS AND THE ZERO PRODUCT PROPERTY If x is either 2 or 3, then y is 0! These are the only two x values for this function that yield this output!

38. TRY IT PROBLEMS Find the roots of the following functions…

39. SOLVING QUADRATICS: COMPLETING THE SQUARE Let’s say that you are trying to factor a quadratic function, and you’re stuck. Trinomial factoring isn’t working, and you’re not sure what else to do. “Completing the square” is a way to create a binomial perfect square that can be easily factored.

40. SOLVING QUADRATICS: COMPLETING THE SQUARE Which factors of -1 add up to 100? …none

41. SOLVING QUADRATICS: COMPLETING THE SQUARE Step 1: Take the constant term, and bring it to the other side of the equals sign.

42. SOLVING QUADRATICS: COMPLETING THE SQUARE Step 2: Add half of b, squared to both sides of the equation.

43. SOLVING QUADRATICS: COMPLETING THE SQUARE Step 3: Represent the left side as a perfect square binomial. Hence, “completing the square.” Which factors of 2500 add up to 100?

44. SOLVING QUADRATICS: COMPLETING THE SQUARE Step 4: Solve for x. Remember: express your solutions in simplest form, not approximated decimal values. Examples

45. TRY IT PROBLEMS Find the solutions of the following quadratic functions using completing the square. Leave your answer in simplest radical form (that is, no decimals).

46. WHEN ALL ELSE FAILS: THE QUADRATIC EQUATION In this video, you will learn the “go to” method for solving quadratic equations. The quadratic equation is a very straightforward method for finding the roots of a quadratic function. Assuming that the equation is in standard form, you can find the roots for any and all quadratic functions, assuming there are roots.

47. WHEN ALL ELSE FAILS: THE QUADRATIC EQUATION [

49. Examples

50. TRY IT PROBLEMS Find the roots of the following functions using the Quadratic Equation. Leave your answer in simplest radical form (that is, no decimals).

51. THE DISCRIMINANT All of the quadratic functions that we have solved so far have come out with two solutions, or in some cases, one solution. However, in this video, we will discuss what happens when we have no solution and how we can determine that using the quadratic equation.

52. THE DISCRIMINANT

54. Tell us the nature of the roots, specifically, how many solutions there are. Two Real solutions No solutions (complex roots) One Real solution

55. Discriminant is positive, so there are 2 solutions

56. Discriminant is zero, so there is 1 solution

57. Discriminant is negative! There are no solutions. Examples

58. TRY IT PROBLEMS Determine the number of solutions that each function has by evaluating the discriminant. You don’t actually need to solve, just determine the number of solutions