WEBVTT 00:20.588 --> 00:28.195 In previous sections, we have studied quadratic equations. In this video, we're going to discuss 00:28.240 --> 00:35.069 quadratic functions and their graphs. A quadratic function is a function that can be written in 00:35.120 --> 00:43.310 the form f of x equals a x squared plus b x plus c where a, b, and c are real numbers 00:43.360 --> 00:49.216 and a is not equal to zero. We're going to begin with the quadratic function 00:49.280 --> 00:57.057 y equals x squared. And if we want to graph this function, we can choose values of x, 00:57.120 --> 01:04.665 calculate the values of y, and then plot those points on our graph. So let's do that. 01:04.720 --> 01:15.376 If x is equal to negative three, then y is equal to negative three squared which is nine. 01:15.440 --> 01:20.748 If x is equal to negative two, then y is equal to negative two squared 01:20.800 --> 01:29.557 which is four. For an x-value of negative one, y will equal negative one squared 01:29.600 --> 01:41.802 which is one. For x equals zero, y is equal to zero squared or zero. For x equal to one, 01:41.840 --> 01:51.645 y is one squared which is one. For x equal to two, y is two squared which is four. 01:51.680 --> 01:59.987 And for x equal to three, y is three squared which is nine. Now let's plot 02:00.020 --> 02:06.594 these as ordered pairs on our graph. We have the ordered pair negative three, 02:06.640 --> 02:13.934 nine. I'm going to use a scaling here of one for each tick mark. So that would mean that this 02:14.000 --> 02:24.111 x-value here is one, two, three, etc. And in the y-values, we will go one, two, three, also. So 02:24.160 --> 02:31.040 negative three, nine would be negative three on the x and positive nine on the y, putting 02:31.051 --> 02:39.460 our ordered pair here. The ordered pair negative two, four will be negative two on the x direction 02:39.520 --> 02:55.376 and four from the y direction. Negative one, one is here. Zero, zero is at the origin. 02:55.440 --> 03:03.717 Then we have one, one so positive one on the x, positive one on the y goes here. 03:03.760 --> 03:16.030 Two, four. Positive two in the x, and up four in the y direction. And three, nine. Three, 03:16.080 --> 03:33.247 nine would be here. Let's connect these points with a smooth curve. 03:33.280 --> 03:39.486 This is the graph of the function y equals x squared. The shape of this function 03:39.520 --> 03:51.966 is called a parabola. You'll notice that this parabola opens in the upward direction. 03:52.000 --> 03:59.540 And it has what we call a turning point. The graph is decreasing here until it gets to x equals zero, 03:59.600 --> 04:09.216 and then it begins to increase. This turning point at zero, zero is called the vertex. 04:09.280 --> 04:16.924 So the vertex is located at zero, zero. In addition, you might notice the symmetry of 04:16.960 --> 04:28.335 this graph. This graph is symmetric about the y-axis. This line here, the line x equals zero, 04:28.400 --> 04:43.550 is called the axis of symmetry. So x equals zero is the axis of symmetry. Throughout this video, 04:43.600 --> 04:50.758 we're going to graph variations of this parent graph, y equals x squared, and we're going to 04:50.800 --> 04:57.798 identify characteristics of the graph such as the direction in which the parabola opens, 04:57.840 --> 05:06.273 where the vertex is located, and where the axis of symmetry is located. 05:06.320 --> 05:14.348 Let's continue graphing variations of the function y equals x squared. Consider the function 05:14.400 --> 05:22.189 f of x equals negative two x squared. We're going to complete this table and graph the 05:22.240 --> 05:30.364 function f of x equals negative two x squared. If x is equal to negative two, 05:30.400 --> 05:38.439 then f of x will be negative two times negative two squared. Negative two squared is four, 05:38.480 --> 05:48.348 and when I multiply four times negative two, I get negative eight. When x is negative one, we have 05:48.400 --> 05:56.924 f of x equal to negative two times negative one squared. Negative one squared is positive one, 05:56.960 --> 06:04.598 and when I multiply positive one times negative two, we get negative two. 06:04.640 --> 06:15.142 When x is zero, we have negative two times zero squared. Anything times zero is zero. When x 06:15.200 --> 06:24.284 is one, f of x is negative two times one squared. That will be negative two times one, or negative 06:24.320 --> 06:33.961 two. And when x is two, f of x is negative two times two squared which will be negative two 06:34.000 --> 06:44.271 times four which is equal to negative eight. Let's plot these as ordered pairs on our graph. 06:44.320 --> 06:50.277 Notice that I already have the graph y equals x squared written here in green 06:50.320 --> 06:57.951 so that we can compare the two graphs after we graph f of x equals negative two x squared. First, 06:58.000 --> 07:04.591 we have the ordered pair negative two, negative eight. Negative two on the x-axis, 07:04.640 --> 07:13.167 and negative eight on the y-axis will be here. Negative one on the x-axis, 07:13.200 --> 07:27.147 and negative two in the y-direction will be here. Zero, zero is at the origin. One, negative two is here. 07:27.181 --> 07:50.604 And two, negative eight is here. Let's connect these ordered pairs or these points with a smooth curve. 07:50.640 --> 08:00.914 This is a sketch of the graph f of x equals negative two x squared. 08:00.960 --> 08:05.219 Now the next part of this question asks us to state the vertex and the 08:05.280 --> 08:12.993 axis of symmetry of the graph of f. The vertex is here at the turning point. 08:13.040 --> 08:29.943 Notice it is the highest point of this graph which is zero, zero. The axis of symmetry is still located here on the y-axis 08:29.977 --> 08:50.931 which is the line x equals zero. Next, we're asked to describe the transformations required to graph f starting from the 08:50.964 --> 09:00.574 graph of y equals x squared. Well, first you may notice that this graph is reflected across the x-axis, so 09:00.640 --> 09:12.986 there is a reflection. Our new graph f of x opens down instead of up. So we say we have a reflection 09:13.040 --> 09:28.735 over the x-axis. You may also notice that the y-values are farther away from the x-axis in 09:28.800 --> 09:35.142 the graph of f of x than they were from the graph y equals x squared. For example, this 09:35.200 --> 09:43.617 ordered pair on y equals x squared was located at one, one. When x is one, the y-value was one. 09:43.680 --> 09:51.625 On our graph of f of x equals negative two x squared, we are now two units away from the x-axis 09:51.680 --> 10:01.134 when x is one. When x is two on the parent function, y equals x squared, our y-value is four. 10:01.200 --> 10:08.742 But on the graph of f of x equals negative two x squared, I am not four units away from the x-axis, 10:08.800 --> 10:16.416 but I am now eight units away from the x-axis in the negative direction. That is a vertical stretch 10:16.480 --> 10:36.203 of a factor of two. So we'll say that we have a vertical stretch by a factor of two. Meaning that each of the 10:36.236 --> 10:48.281 y-values are two times farther away from the x-axis than they were from the parent graph. 10:48.320 --> 10:55.455 In this next example, let's consider the two functions f of x equals x squared and 10:55.520 --> 11:04.030 g of x equals x squared minus two. We're going to begin by graphing f and g on the same axis. 11:04.080 --> 11:09.703 Notice, we've already discussed the graph of y equals x squared, and so I have used that 11:09.760 --> 11:16.042 information to complete the table here for f of x equals x squared, and its corresponding graph 11:16.080 --> 11:24.017 is here on our coordinate plane. Now let's use g of x equals x squared minus two to determine the 11:24.080 --> 11:33.560 values the y-values for these corresponding x-values. When x is equal to negative two, 11:33.600 --> 11:42.102 g will equal negative two squared minus two. Negative two squared is four, and four minus 11:42.160 --> 11:53.880 two is two. When x is negative one, g of x will be negative one squared minus two. 11:53.920 --> 12:02.255 Negative one squared is positive one, and one minus two is negative one. 12:02.320 --> 12:11.464 When x is zero, g of zero will be zero squared, which is zero, minus two, 12:11.520 --> 12:22.509 giving us a value of negative two. When x is one, g of one will give us one squared minus two 12:22.560 --> 12:30.917 which will be one minus two, or negative one. And when x is two, we'll have g of two 12:30.960 --> 12:41.862 equal to two squared minus two which is four minus two, or two. Let's plot these ordered pairs 12:41.920 --> 12:52.906 for the graph of g. We have the ordered pair negative two, two. Negative two, two. 12:52.960 --> 13:14.895 We have the ordered pair negative one, negative one. That's here. Zero, negative two. One, negative one. 13:14.960 --> 13:36.182 And two, two. And now we'll connect these points with a smooth curve. The next part 13:36.240 --> 13:45.392 of our problem asks us to state the vertex and the axis of symmetry of the graphs of f and g. 13:45.440 --> 13:55.001 I'll put f here and g here. Well, for the graph of f, that was our parent function f 13:55.040 --> 14:03.877 of x is equal to x squared, we notice that the vertex is at zero, zero. 14:03.920 --> 14:19.726 And for g notice that the vertex is here at zero, negative two. The axis of symmetry for both of these graphs 14:19.760 --> 14:34.341 is the y-axis, or the line x equals zero. Both graphs are symmetric about the y-axis. 14:34.400 --> 14:41.314 The last question asks us to describe the graph of g as a transformation of the graph of 14:41.360 --> 14:52.659 f. Well, first let me label g on this graph. This is g of x equal to x squared minus two. 14:52.720 --> 15:01.935 And our green graph was f of x equals x squared. Notice that each of these points of the red graph, 15:02.000 --> 15:12.445 g of x, are two units lower. The y-value is two units less than the y-value of the parent graph. 15:12.480 --> 15:21.721 This y-value is at one. This y-value is two units lower. The same here and here. In fact, for any 15:21.760 --> 15:29.229 point that's on the parabola the red graph is two units lower than the corresponding points on the 15:29.280 --> 16:01.227 green graph. So we say that the transformation of g is that it is a vertical shift down two units. 16:01.280 --> 16:06.266 All quadratic functions can be graphed using transformations of the graph 16:06.320 --> 16:13.807 y equals x squared. In order to precisely graph a quadratic function, you need a minimum of three 16:13.840 --> 16:19.712 points, one of which is the vertex. You can see here that I have graphed our equation, 16:19.760 --> 16:27.487 or our function, y equals x squared using three points, one of which is the vertex. 16:27.520 --> 16:34.894 Now let's take a look at this example. We're going to graph the function f of x equals one-fourth 16:34.960 --> 16:41.401 times x minus four squared plus three. And we're going to graph it, instead of using a table 16:41.440 --> 16:48.908 of values, we're going to use transformations of this function y equals x squared. So let's 16:48.960 --> 16:56.583 recall what we have learned previously about transformations. When we add or subtract inside 16:56.640 --> 17:04.924 of grouping symbols, the transformation is a horizontal shift left or right. So this minus four 17:04.960 --> 17:12.599 inside grouping symbols will require us to shift our graph to the right four units. 17:12.640 --> 17:18.338 The one-fourth that is multiplied outside of these parentheses is either a vertical stretch 17:18.400 --> 17:26.679 or a vertical compression. Because this number is a value between zero and one, less than one, 17:26.720 --> 17:34.654 then this is a vertical compression. So all of our y-values will be multiplied by one-fourth. 17:34.720 --> 17:41.961 They'll be a fourth of the distance from the x-axis as our parent graph originally has. 17:42.000 --> 17:47.400 And then our plus three at the end of the function, or outside of the grouping symbols, 17:47.440 --> 17:56.910 is a vertical shift. Adding three here will be a vertical shift up three. So we'll begin 17:56.960 --> 18:03.883 with the points that we have identified here, and we will apply these three transformations. 18:03.920 --> 18:12.525 Let's start with our vertex. We're going to shift our vertex to the right four units. Multiplying 18:12.560 --> 18:23.002 this y-value by one-fourth still gives us a value of zero, and then adding three puts us here. 18:23.040 --> 18:34.047 So this ordered pair is four, three. Now let's move this ordered pair from our original graph, 18:34.080 --> 18:44.023 which was at negative two, four. We'll begin with the horizontal shift to the right four units. 18:44.080 --> 18:52.498 Next, we'll multiply this y-value by one-fourth. The y-value here on the parent graph is four. When 18:52.560 --> 19:02.742 we multiply that by one-fourth, the new y-value is one. And then we have a vertical shift of positive 19:02.800 --> 19:17.056 three, so this will shift this point back up three units, putting us here at our new point two, four. 19:17.120 --> 19:23.629 And our third point, which was originally at two, four, will also follow these transformations. 19:23.680 --> 19:31.938 We'll move this point to the right four. We'll multiply the y-value by one-fourth, 19:32.000 --> 19:40.847 which brings us to one, and then we will add three for our vertical shift, 19:40.880 --> 20:00.033 bringing us to six, four. And now we'll connect these three points with a smooth curve. 20:00.080 --> 20:09.142 This is the graph of f of x equals one-fourth x minus four squared plus three. Now let's 20:09.200 --> 20:25.958 state the vertex and the axis of symmetry of the graph of f. The vertex is here at four, three. 20:26.000 --> 20:38.604 And the axis of symmetry runs through the vertex. It is the line x equals four. 20:38.640 --> 20:46.112 The graph of a function written in this form f of x equals a times x minus h squared plus 20:46.160 --> 20:55.288 k is a parabola that has the following features. The vertex will be h comma k. 20:55.321 --> 21:05.698 If a is greater than zero, a positive value, then the parabola will open upward. If a is less than 21:05.760 --> 21:13.873 zero, the parabola will open downward. The axis of symmetry in either case will be the vertical 21:13.920 --> 21:22.515 line x equals h. Here you can see I've graphed a parabola that opens downward on this side, 21:22.560 --> 21:32.258 where a is less than zero. Here's the vertex at h, k. And the axis of symmetry is x equals h. 21:32.320 --> 21:41.234 And here is a parabola that opens upwards, so a is greater than zero. The vertex is h comma k, 21:41.280 --> 21:51.310 and the axis of symmetry is x equals h. Let's continue with some more examples. 21:51.360 --> 21:58.918 Let's graph the function f of x equals negative x plus one squared plus five. 21:58.960 --> 22:03.623 Using what we just learned about the vertex and axis of symmetry, 22:03.680 --> 22:16.335 from this equation, we can identify that the vertex is located at negative one, five. 22:16.400 --> 22:30.349 And the axis of symmetry is the vertical line that runs through the x-value negative one. So we have an axis of symmetry 22:30.400 --> 22:37.723 of the line x equals negative one. To find another point that's on this parabola, 22:37.760 --> 22:45.298 let's find the y-intercept. To find the y-intercept, we set x equal to zero. So 22:45.360 --> 22:57.243 we're going to find f of zero. This gives us negative zero plus one squared plus five. 22:57.280 --> 23:03.883 This will be one squared multiplied by negative one, which is negative one, 23:03.920 --> 23:17.229 plus five, giving us four. So f of zero is equal to four. This is our y-intercept. 23:17.280 --> 23:24.203 Recognizing that our axis of symmetry is here at x equals negative one means that there will be 23:24.240 --> 23:31.944 a point on the other side of the axis of symmetry one unit from the axis of symmetry with the same 23:32.000 --> 23:46.993 y-value. Now we have three points, and we can sketch the graph of this function. 23:47.040 --> 23:54.367 The next part of the problem asks us to state the vertex and the axis of symmetry of the graph 23:54.400 --> 24:05.478 of f which we began with. We noticed that the vertex was located at negative one, five, 24:05.520 --> 24:17.840 and the axis of symmetry is the vertical line x equals negative one.