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3. Axis of symmetry: The vertical line that divides a parabola into two  symmetrical halves

4. Maximum: The y-value of the vertex if a < 0 and the parabola opens  downward Minimum: The y-value of the vertex if a > 0 and the parabola opens  upward Parabola: The curve result of graphing a quadratic equation Max (+ a) (- a) Min

5. Quadratic Equation: An equation that can be written in the standard  form ax2 + bx + c = 0. Where a, b and c are real numbers and a does  not = 0. Quadratic Function: Any function that can be written in the form y =  ax2 + bx + c. Where a, b and c are real numbers and a does not equal  0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero.

7. Any quadratic function that can be written in the form y = ax2 + bx + c (standard form) Where a, b, and c are real numbers and a ≠ 0 Examples Question: Is y = 7x + 9x2- 4 written in standard form? Answer: No Question: Is y = 0.5x2 + 7x written in standard form? Answer: Yes

9. A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0. The form ax2 + bx + c = 0 is called the standard form of the quadratic equation. The standard form is not unique. For example, x2 - x + 1 = 0 can be written as the equivalent equation -x2 + x - 1 = 0. Also, 4x2 - 2x + 2 = 0 can be written as the equivalent equation 2x2 - x + 1 = 0. Why is this equivalent?

10. Practice writing quadratic equations in standard form: (Reduce if possible.) Write 2x2 = x + 4 in standard form: 2x2 - x - 4 = 0

11. Write 3x = -x2 + 7 in standard form: x2 + 3x - 7 = 0

12. Write 6x2 - 6x = 12 in standard form: x2 - x - 2 = 0

13. Write 3x - 2 = 5x in standard form: Not a quadratic equation

14. Characteristics of Quadratic Functions 1. Standard form is y = ax2 + bx + c, where a ≠ 0.

15. 2. The graph of a quadratic is a parabola, a u-shaped figure. 3. The parabola will open upward or downward. ← upward downward →

16. 4. A parabola that opens upward contains a vertex that is a  minimum point. A parabola that opens downward contains a  vertex that is a maximum point. vertex vertex

17. 5. The domain of a quadratic function is all real numbers.

18. 6. To determine the range of a quadratic function, ask yourself two questions:  Is the vertex a minimum or maximum?    What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is -6 to

19. If the vertex is a maximum, then the range is all real numbers less  than or equal to the y-value. The range of this quadratic is to 10

20. 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of  symmetry is always a vertical line of the form x=2

21. 8. The x-intercepts are the points at which a parabola  intersects the x-axis. These points are also known as  zeroes, roots or solutions and solution sets. Each  quadratic function will have two, one or no real x-intercepts.

22. 1 True or False: The vertex is the highest or  lowest value on the parabola. True False

23. 2 If a parabola opens upward then... A a>0 B a<0 C a=0

24. 3 The vertical line that divides a parabola  into two symmetrical halves is called... A discriminant B perfect square C axis of symmetry D vertex E slice

25. 4 What is the equation of the axis of symmetry of the parabola shown in the diagram  below? 20 18 16 14 12 10 8 6 4 2 A x = -0.5 B x = 2 C x = 4.5 8 9 10 5 6 7 1 2 3 4 5 D x = 13

26. 5 The height, y, of a ball tossed into the air can be represented by the equation  y = −x2 + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola? A y = 5 B y = -5 C x = 5 D x = -5

27. 6 What are the vertex and axis of symmetry of the parabola shown in the  diagram below? A vertex: (1,−4); axis of symmetry: x = 1 B vertex: (1,−4); axis of symmetry: x = −4 C vertex: (−4,1); axis of symmetry: x = 1 D vertex: (−4,1); axis of symmetry: x = −4

28. 7 The equation y = x2 + 3x − 18 is graphed on the set of axes below. Based on this graph, what are the roots  of the equation x2 + 3x − 18 = 0? A −3 and 6 B 0 and −18 C 3 and −6 D 3 and −18

29. 8 The equation y = − x2 − 2x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation − x2 − 2x + 8 = 0? A 8 and 0 B 2 and -4 C 9 and -1 D 4 and -2

31. Graph by Following Six Steps: Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect

32. Step 1 - Find the Axis of Symmetry What is the Axis of Symmetry? The line that runs down  the center of a parabola.  This line divides the  graph into two perfect  halves. Pull Axis of Symmetry

33. Step 1 - Find the Axis of Symmetry Graph y = 3x2 – 6x + 1 Formula: a = 3 b = -6 x = - (- 6) = 6 = 1 2(3) 6 The axis of  symmetry is x = 1.

34. Step 2 - Find the vertex by substituting  the value of x (the axis of symmetry)  into the equation to get y. y = 3x2 - 6x + 1 a = 3, b = -6 and c = 1 y = 3(1)2 + -6(1) + 1 y = 3 - 6 + 1 y = -2 Vertex = (1 , -2)

35. Step 3 - Find y intercept. What is the y-intercept? The point where the line  passes through the y-axis. This occurs  when the x-value is 0. Pull y- intercept

36. Graph y = 3x2 – 6x + 1 Step 3 - Find y- intercept. The y- intercept is always the  c value, because x = 0. y = ax2 + bx + c c = 1 y = 3x2 – 6x + 1 The y-intercept is 1 and the graph passes through (0,1).

37. Graph y = 3x2 – 6x + 1 Step 4 - Find two points Choose different values of x  and plug in to find points. Let's pick x = -1 and x = -2 y = 3x2 – 6x + 1 y = 3(-1)2 – 6(-1) + 1 y = 3 +6 + 1 y = 10 (-1,10)

38. Step 4 - Find two points Graph y = 3x2 – 6x + 1 y = 3x2 - 6x + 1 y = 3(-2)2 - 6(-2) + 1 y = 3(4) + 12 + 1 y = 25 (-2,25)

39. Step 5 - Graph the axis of symmetry,  the vertex, the point containing the y-intercept and two other points.

40. Step 6 - Reflect the points across  the axis of symmetry. Connect the  points with a smooth curve. (4,25)

41. 9 What is the axis of symmetry for y = x2 + 2x - 3 (Step 1)? A 1 B -1

42. 10 What is the vertex for y = x2 + 2x - 3 (Step 2)? A (-1,-4) B (1,-4) C (-1,4)

43. 11 What is the y-intercept for y = x2 + 2x - 3 (Step 3)? A -3 B 3

44. 12 What is an equation of the axis of symmetry of the parabola represented by  y = −x2 + 6x − 4? A x = 3 B y = 3 C x = 6 D y = 6

45. Graph

46. Graph

47. Graph

48. On the set of axes below, solve the following system of equations graphically for all values of x and y . y = -x2 - 4x + 12 y = -2x + 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

49. On the set of axes below, solve the following system of equations graphically for  all values of x and y. y = x2 − 6x + 1 y + 2x = 6

50. 13 The graphs of the equations y = x2 + 4x - 1 and y + 3 = x are drawn on the same set of axes. At which point do the graphs intersect? A (1,4) B (1,–2) C (–2,1) D (–2,–5)