1. Presented by Dr. Del Ferster

2.  We’ll spend a bit of time looking at some “test-type” problems.  We’ll re-visit quadratic functions. This time, we’ll emphasize the graphs of quadratics. ◦ We’ll look at vertex and the intercepts.  We’ll spend a bit of time on the graphs of linear functions. ◦ We’ll consider intercepts, and slope intercept form.  Then, we’ll look at some problems that model both kinds of functions (quadratic and linear).

3. Practice Problems Graphing Quadratic Functions and Linear Functions

4.  1. y=(x – 2)^2 -2, so B  2. Vertex is at (-3, 4), so C  3. C  4.y-int. (0,3): x-int. (4,0), so A  5., so C  6. (0, 16), so D 

5.  7. (4, 0) or (2, 0), so C  8.Vertex at (3,-2), opens up, so C  9A. 

6.  9B. m=-2  9C.  9D.(0,4) or 4  9E.(2,0) or 2  10.  11A.(0,-5) or -5  11B. 5 or 1, or alternately (5, 0) or (1, 0)  (3, -2) is a minimum, so C  10.  11A.(0, 10)  11B.5 or 1, or alternately (5, 0) or (1, 0)  11C.(3, -8)

7.  11C.(3,-4)  11D.

8. Quadratic Functions Graphs, Forms, and Intercepts

10.  The graph of any quadratic function is called a parabola.  Parabolas are shaped like cups, as shown in the graphs on the next slide.  If the coefficient of x 2 is positive, the parabola opens upward; otherwise, the parabola opens downward.  The vertex (or turning point) is the minimum or maximum point.

12.  We’ll consider several forms, each tends to have some information that we can retrieve easily.  1.VERTEX FORM: In this form, we’re able to easily identify the vertex of the parabola. (h,k) y  a(x  h) 2  k, a  0

13.  2.STANDARD FORM: This form works well if you wish to apply the quadratic formula to find the zeros of the function. (ZEROS of a function are the places where the graph of the function crosses the x axis.

14.  3.FACTORED FORM: This form makes it easy to find the zeros of the function. We just use the factors The zeros are at

15.  We often seek to find the following information about a quadratic function. ◦ The y intercept ◦ The zeros ◦ The vertex ◦ Whether the graph opens upward or downward ◦ The maximum or minimum value of the function (depending on how the graph opens)

16.  Let’s look at these ideas one at a time. ◦ The y intercept ◦ This is relatively easy to find, if we realize that the location where a graph crosses the y axis, it’s x value is ZERO. ◦ So, to find a y intercept, just “plug 0 in for x”.

17.  Let’s look at these ideas one at a time. ◦ The zeros ◦ Depending on the form, these can be easy or tougher to find. ZEROS are what most books now call the places where the graph crosses the x axis. You might have once called these x intercepts. ◦ So, to find a zero, just “set y equal to 0, and get ready to solve for x”.

18.  Let’s look at these ideas one at a time. ◦ The vertex ◦ This is the “turning point” of the curve. It’s also where the graph reaches its maximum or minimum value (depending on whether it opens downward or upward. ◦ In vertex form, this is (h,k).

19.  If by chance, your function isn’t in vertex form, but maybe instead is in standard form  In this case, the vertex has an x coordinate of

20.  Let’s look at these ideas one at a time. ◦ Whether the graph opens upward or downward ◦ This is easy to determine. We simply look at the coefficient on the square term.  If it’s positive, the graph opens UPWARD  If it’s negative, the graph opens DOWNWARD

21.  Let’s look at these ideas one at a time. ◦ The maximum or minimum value of the function (depending on how the graph opens) ◦ This is easy to determine, and it depends on whether the graph opens upward or downward. In either case, the max/min value of the function is the y value at the vertex.  If the graph opens UPWARD we have a minimum  If the graph opens DOWNWARD we have a maximum.

22.  For the function  A.Find the y intercept  B.Find the zeros  C.Find the vertex  D.Determine whether the graph opens upward or downward  E.Find the maximum or minimum value of the function.

23.  To find the y intercept:  Plug 0 in for x So, our y intercept is (0, -6)

24.  To find the zeros:  Set y equal to zero and solve for x So, our zeros are at 3 or -1

25.  This is easy to do, since we’re given vertex form:  The vertex is (h, k) So, our vertex is (1, -8)

26.  To the opening direction, just look at the number in front.  In this case it’s a 2 (positive) So, our graph opens UPWARD

27.  In this case, since the graph opens UPWARD, we have a minimum  It occurs at the vertex (which is at (1, -8) So, our function has a minimum value of -8

28. Y intercept (0, -6) zeros are at 3 or -1 vertex is (1, -8) graph opens UPWARD So, our function has a minimum value of -8

29. A closer look at Quadratic Functions VERTEX FORM STANDARD FORM

30.  We’ll look at the process of converting a quadratic function from standard form to vertex form.  Then, just for kicks, we’ll look at the process for converting a quadratic function from vertex form to standard form.  It has been my experience, that students usually prefer vertex form, especially those students who are good at shifts.

31.  A couple of thoughts: ◦ If the graph of the parabola crosses the x axis at ONLY one point, that point is the vertex. ◦ If the graph of the parabola crosses the x axis at 2 points, the vertex falls in the middle of these two x values.  In this case, we can find the zeros, then “average them” to determine the x value of the vertex. ◦ If there are no zeros, YIKES…finding the vertex in this case is a bit tougher.

32.  Although it’s not one of my favorite mathematical ideas, there is a “formula” that allows us to locate the vertex, if we have a quadratic function in standard form.  Recall that standard form is: ◦ The vertex is located at the point

33. The general form: The formula for vertex: To find the x coordinate of the vertex: To find the y coordinate of the vertex Plug the above x value into the function, and get a y value to go along with it The general form: The formula for vertex: To find the x coordinate of the vertex: To find the y coordinate of vertex

34.  Find the vertex, then sketch the graph: Plug the 3 into the function, we get a y value of 9-18+20 or y = 11 The Vertex

36.  Recall that vertex form looks like this:  While standard form looks like this:

37.  Really, it’s not hard to do at all; we just “multiply out” the expression, using our “FOIL” skills, and group the like terms.  Let’s try one!  Convert the function to standard form:

38.  Unfortunately, this procedure isn’t quite as simple. However, there are 2 ways that we could proceed. ◦ 1. We can complete the square ◦ 2. We can make use of the vertex formula.  Let’s take a look at an example for each!

39.  First, we’ll look at an “easy one”—where the value of a is 1.  We’ll work it through on the board, then I’ll reveal the steps here on the PowerPoint.  Rewrite the function in vertex form:

40.  Steps in the procedure: ◦ 1. take half of b ◦ 2. square the result ◦ 3. ADD (to complete the square) this amount to the function and subtract this amount from the function ( aha…thus we’ve added ZERO…and not changed the value of the function) ◦ 4. Rewrite as a binomial squared, and we’re there. ◦ Half square ADD Rewrite

41.  First, group the x terms together, leave some space.  Now, take half of the 10, square it, and add it AND subtract it.  Rewrite the first 3 terms as a binomial squared  So our vertex is at  (-5, -37)

42.  In this case, we first FACTOR out the value of a to get a look at a coefficient of 1 on our x squared term….it’s a bit messier, and we must be careful when we do the subtract part.  Let’s do one together.  Rewrite the function in vertex form:

43. Vertex is (4, -60)

44.  Recall from a previous slide, that the parabola  has its vertex at:  We can use this idea to help to transform the equation into vertex form, also.

45.  Rewrite the function in vertex form, and identify the vertex.  The x coordinate for the vertex is

46.  Now, we plug that x value into the function, to find the y value for the vertex point.  So the vertex is (-2, -18) and the form becomes

47.  Vertex Form: ◦ Vertex is at (h, k)  Standard Form:  The value of a determines stretch or compress.  The value of a determines whether the graph opens up or down.  The value of a determines the concavity of the graph.

48.  1.Change the given equation to vertex form  2.Determine the vertex.  3.Determine the y intercept.  4.Sketch the graph.

49. Vertex Form Vertex Y intercept

50. The graph

51. Linear Functions Graphs, Slope, and Intercepts

52. Y X An Intercept is the coordinate where a line crosses the x or y axis

53. The X intercept is the x coordinate (where a line crosses the x axis). Y X (0, 2 ) ( 3,0) The Y intercept is the y coordinate (where a line crosses the y axis).

54. Y X (0, 2 ) ( 3,0)

55. Y X

56. 1stt chart X Y = 2x - 6 Y X Graph Linear Eq.

57. 1stMake x-y table 2ndSet x = 0 and solve for y X Y = 2x - 6 0 -6 Y X (0,-6) Graph Linear Eq.

58. 1stMake x-y table 2ndSet x = 0 and solve for y 3rdSet y = 0 and solve for x X Y = 2x - 6 0 -6 30 Y X (0,-6) (3,0) Graph Linear Eq.

59. 1stMake x-y table 2ndSet x = 0 and solve for y 3rdSet y = 0 and solve for x 4th Plot these 2 points and draw line X Y = 2x - 6 0 -6 30 Y X (0,-6) (3,0) Graph Linear Eq.

60. 1stMake x-y table 2ndSet x = 0 and solve for y 3rdSet y = 0 and solve for x 4th Plot these 2 points and draw line 5thUse 3rd point to check (this is totally optional…in fact, I don’t usually do it.) X Y = 2x - 6 0 -6 30 4 2 Y X (0,-6) (3,0) (4,2) Graph Linear Eq.

61. Y X This line has a y value of 4 for any x-value. It’s equation is y = 4 (meaning y always equals 4)

62. Y X This line has a x value of 1 for any y-value. It’s equation is x = 1 (meaning x always equals 1)

63. Y X x = 1

64. Y X y = 3

65. Slope is a measure of STEEPNESS

67. How much does this line rise? How much does it run? (3,2) (6,4) (0,0) 1 2 3 1 2 4 3 56 4

68. How much does this line rise? How much does it run? (3,2) (6,4) (0,0) 1 2 3 1 2 4 3 56 4

69. (3,2) (6,4) (0,0) 1 2 3 1 2 4 3 56 4 x1y1x1y1 x2y2x2y2

70. ZERO Slope -- Horizontal Positive Slope Is Up Negative Slope Is Down Undefined Slope-- Vertical

71. There are 3 Forms of Line Equations Standard Form: ax+by=c Slope Intercept Form: y=mx+b Point-Slope Form y-y 1 =m(x-x 1 ) (We’ll make use of this next class) All 3 describe the line completely

72. Slope Intercept Form: y=mx+b JUST SOLVE FOR Y

73. The great thing about this form is b is the y-intercept. This makes graphing a line incredibly easy. Check it out. If (0,1) The y intercept is +1 Almost a free point on graph

74. All you have to do now is use the slope to rise and run from the intercept & connect the points. (0,1) Rise 2 and Run 3 from the y-intercept & connect points.

75. All you have to do now is use the slope to rise and run from the intercept & connect the points. (0,1) Rise -2 and Run 3 from the y-intercept & connect points.

76. Now it is This line has an y intercept of -2 and rises 5 and runs 2. Solution Steps to Solve for y: Divide by 2

77. 1. Solve for y: 2.Y-Intercept is 1st Point. 3.From the y-intercept Rise 5 and run 2 for Second Point. 4.Connect Points with line. Graph (0,-2) 5 2

78. Now it is easy to graph (0,-2)

79. 1.Make sure equation is in y=mx+b form 2.Plot b(y-intercept) on graph (0,b) 3.From b, Rise and Run according to the slope to plot 2nd point. 4.Check sign of slope visually

80.  Thanks for your attention and participation. ◦ You have my utmost respect for working hard all day with your kids, and still hanging in there for what we’ve done!  Hang in there, only a week till Thanksgiving, and we all have much to be thankful for  If I can help in any way, don’t hesitate to shoot me an email, or give me a call.