Graphing Quadratic Functions in General Form — Transcript

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1. 0:01This video will review the quadratic function.
2. 0:04More specifically, it will review how to  graph a quadratic function in general form.
3. 0:09Let's take a look at some of the properties  of a quadratic function in general form. The
4. 0:15graph of a quadratic function, f of x equals  a times x squared plus b times x plus c,
5. 0:20is called a parabola. It is always a cup-shaped or  U-shaped curve. The graph of a quadratic function,
6. 0:27f of x equals a times x squared plus b  times x plus c, is called a parabola. It
7. 0:33is always a cup-shaped or U-shaped curve.  It opens upward if a is greater than zero,
8. 0:39or opens downward if a is less than zero. The  vertical line x equals negative b over two times a
9. 0:46is the line of symmetry. It has a turning point  or a vertex at a point where the x-coordinate
10. 0:54is negative b over two times a and the  y-coordinate is f of negative b over two times a.
11. 1:00Let's take a look at the graph of these important  features. First off, the parabola is in blue.
12. 1:06Notice how it opens upward; therefore, we can  conclude in this case that a would be greater
13. 1:11than zero. The line of symmetry is in red. Notice  how the reason it's called the line of symmetry is
14. 1:18if you were to fold the parabola across that line,  it would match up perfectly with the other half.
15. 1:25The vertex is the only point that is on the  graph as well as on the line of symmetry.
16. 1:31It will either be a high point  or a low point on the graph,
17. 1:34and in this case, since it opens  upward, the vertex is a low point.
18. 1:39Two other important components are the  x-intercepts and the y-intercepts. As we
19. 1:43know from our previous studies, x-intercepts are  where the graphs cross the x-axis. Some parabolas
20. 1:50may not have x-intercepts, and of course, the  y-intercept is where the graph crosses the y-axis.
21. 1:58Let's explore the graph of a quadratic in  general form. I'm going to go to this website.
22. 2:16Now, this equation is in general form, and  what these sliders allow you to do is adjust
23. 2:20the values of a, b, and c. We're going to look  at what happens when we change the value of a.
24. 2:27Notice how as a increases, the green  parabola gets narrower and narrower,
25. 2:34but it does still continue to open upward.  And as soon as I slide this so the value of
26. 2:39a is less than one, it becomes wider. And then  when it reaches a negative value, it does open
27. 2:46downward, so all the values of a that are  negative, the parabola will open downward.
28. 2:53I'll leave this for you to play with on your  own time, but let's take a quick look at what
29. 2:57the value of c does. As I increase the value  of c, notice how it's a vertical shift upward.
30. 3:03As I decrease it, it's a vertical shift downward.  And the other connection is that the value of c
31. 3:09is actually where the graph crosses  the y-axis or the y-intercept.
32. 3:15Let's go back to our presentation.
33. 3:18Let's take a look at our own examples. Graph f of  x equals x squared minus four x minus five. Find
34. 3:25the equation of the axis of symmetry or line of  symmetry, the vertex, and the x-intercepts. Okay,
35. 3:31so remember to find the line of symmetry we have  an equation for that, and that would be x equals
36. 3:39negative b over two times a. The first thing I  like to do is identify the values of a, b, and c.
37. 3:46Those are the coefficients.  So a would be equal to one,
38. 3:50b would be equal to negative four,  and c is equal to negative five.
39. 3:56So doing our substitutions into our formula for  the axis of symmetry, we would have negative
40. 4:04negative four all over two times one, which would  give us x equals two for our line of symmetry.
41. 4:13Remember, when we go to find the vertex,  the value of x for the line of symmetry
42. 4:18also gives us the x-coordinate of  the vertex. In order to find the
43. 4:23y-coordinate, we have to substitute  two into the original function,
44. 4:27but f of two is equal to negative nine. Therefore,  our vertex is two negative nine. Recall,
45. 4:34to find the x-intercepts of any function, we  have to set y equal to zero and solve for x.
46. 4:40So we would have zero equals x squared minus  four x minus five. Luckily, this is factorable.
47. 4:53The solutions to this quadratic are x equals five  and x equals negative one. These are also our
48. 4:59x-intercepts, so we can write our x-intercepts  as the two points five zero and negative one
49. 5:08zero. Let's go ahead and put all  these pieces together in a graph.
50. 5:13Here it is in red. We have our  line of symmetry x equals two,
51. 5:19x-intercept of positive five,  x-intercept of negative one,
52. 5:26our vertex with the coordinates two negative  nine. Notice how with these three points we
53. 5:32could actually make a nice graph just by  hand without the use of any technology.
54. 5:37Let's take a look at one more example. Given f  of x equals negative two x squared plus ten x
55. 5:43minus seven, find the equation of the  axis of symmetry again or line of symmetry
56. 5:49and the vertex, then graph with the  help of the graphing calculator.
57. 5:54Going back to our equation  for the axis of symmetry.
58. 6:00The
59. 6:01first thing we need to do is identify the values  of a and b. Our a value would be negative two,
60. 6:08b is ten. We don't need c, but c would be  negative seven. Okay, let's do the substitution
61. 6:16and see what we get. Negative negative  b would be negative ten over two times
62. 6:24our a value of negative two. It's like  we'd have ten fourths or five halves.
63. 6:32The line of symmetry is x equals five  halves or if we want two point five.
64. 6:40Of course, the vertex again, we know the  x-coordinate of the vertex has to be five halves.
65. 6:45In order to find the y-coordinate of the vertex,  we have to do the substitution into the original
66. 6:53equation with x equal to five halves,
67. 6:56and you can go ahead and do that,  but I came up with eleven halves.
68. 7:00Notice how this problem does not ask us  to find the x-intercepts of this parabola,
69. 7:06and the reason it doesn't actually is because if  you set y equal to zero, the quadratic equation
70. 7:10is not factorable. So let's go ahead and get our  graphing calculators out and use some technology.
71. 7:18If you hit y equals, I've already typed in the  function. Now if I hit graph, first thing I
72. 7:24want you to notice is it does open downward, and  the reason it opens downward again is because a
73. 7:28is negative. However, it's very difficult to get  additional points on the graph from this screen,
74. 7:34so it's often more helpful if you hit second graph  and pick some points from the table. For example,
75. 7:40I could pick the points one one,  two, and maybe zero negative seven.
76. 7:52Of course, there were a bunch of other points  that we could pick. I just decided to pick
77. 7:57these three. Let's see if we can graph this  and put all these key components together.
78. 8:06First thing we noticed here is our  axis of symmetry or line of symmetry,
79. 8:10and we said that was equal to x  equals two point five or five halves.
80. 8:16Here's the vertex where we found the  x-coordinate to be five halves, of course,
81. 8:21and the y-coordinate to be eleven halves. There  it is. And we found a few other points on the
82. 8:27graph. I believe we found the point one one, zero  negative seven, and I think we found two five.
83. 8:37And notice how with the axis of symmetry, if you  had these points on the left side of the axis,
84. 8:43you could easily find their mirror images on the  right. Meaning if there is a point here two five,
85. 8:48there has to be another point that's a mirror  image of that on the other side of the axis
86. 8:52of symmetry. Same thing with this point; there  has to be another point somewhere over here that
87. 8:58helps you find additional points, and that's why  it's so important to find the axis of symmetry.
88. 9:03I hope that helps you review how to graph  quadratic functions in general form.
89. 9:08Have a great day!