05 - Graphing Parabolas - Opening Up and Down (Quadratic Equations) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . I'm Jason with math and | |
| 00:02 | science dot com . Today we're going to cover the | |
| 00:04 | concept of graphing parabolas problem . Open up , opening | |
| 00:07 | upward and opening downward . So we're inching our way | |
| 00:11 | forward , learning and dissecting the equations of these parabolas | |
| 00:14 | . Now we're going to concentrate on what makes a | |
| 00:16 | problem open upward like a smiley face or if it | |
| 00:19 | opens downward like a frowny face , what makes that | |
| 00:21 | happen . And so in this equation or in this | |
| 00:23 | board , this is what we did in the last | |
| 00:25 | lesson and I just left it up here . I | |
| 00:26 | haven't really made any changes , but we talked about | |
| 00:28 | the concept of the basic parabola . F of X | |
| 00:31 | is equal to X square . And we did a | |
| 00:32 | quick little table of values and we plotted that and | |
| 00:35 | we drew the general shape of the problem . And | |
| 00:37 | this is an example of a parabola that opens upward | |
| 00:40 | because obviously it opens upward and the opposite of that | |
| 00:43 | would be a problem that opens downward . So I'm | |
| 00:45 | going to focus on this equation now , because the | |
| 00:47 | basic parabola is f of X is equal to x | |
| 00:50 | square . It's the most basic one you can have | |
| 00:52 | . Let's start to talk a little bit about the | |
| 00:54 | more general form of the parabola here . So a | |
| 00:57 | more general form more , I'm not going to say | |
| 01:00 | it's the complete general form , but it's more general | |
| 01:05 | parabola than what we have here is the following . | |
| 01:11 | And uh the more general form of the problem is | |
| 01:14 | Y equals a times X squared . So the only | |
| 01:18 | thing that we've done is we've added something called A | |
| 01:20 | in front of the X squared and we say that | |
| 01:23 | A is positive or negative , but it cannot equal | |
| 01:30 | zero . In other words , examples here , some | |
| 01:33 | examples , some real examples , you can have A | |
| 01:36 | is equal to two X squared A would be to | |
| 01:39 | you could have A is equal to four X squared | |
| 01:41 | . You could have a is equal to 1.5 X | |
| 01:44 | squared . You can have A is equal to three | |
| 01:45 | quarters 3/4 X squared . It doesn't matter if it's | |
| 01:48 | a decimal fraction , it just has to be a | |
| 01:50 | number bigger than zero or it could be uh smaller | |
| 01:53 | than zero , it could be negative for X squared | |
| 01:56 | , it could be negative two X squared . It | |
| 01:58 | could be negative 19 X squared . It could be | |
| 02:00 | negative one half X squared . But it can't be | |
| 02:03 | zero because if you have a zero here then the | |
| 02:05 | whole thing goes to zero and why is equal to | |
| 02:07 | zero and why is equal to zero ? Is a | |
| 02:09 | horizontal line in the on the , along the X | |
| 02:12 | axis there . It's not a problem at all . | |
| 02:14 | So A has to be something other than zero . | |
| 02:17 | Otherwise the thing just kills it and it's not a | |
| 02:20 | problem at all . So you might say how does | |
| 02:22 | that go with what we had here ? Well , | |
| 02:24 | if you think about the general form of a problem | |
| 02:26 | being a X square , then I have an invisible | |
| 02:29 | one right here . So in the case of the | |
| 02:31 | basic problem , which is what I told you , | |
| 02:33 | I want you to burn in your mind A is | |
| 02:35 | just equal to one . So when you have a | |
| 02:37 | is equal to one , you have that very basic | |
| 02:39 | kind of like the most central problem that we have | |
| 02:42 | and anything other than A is going to slightly change | |
| 02:44 | the shape of the problem . And the most important | |
| 02:47 | thing about A . That we want to talk about | |
| 02:49 | this concept of A is the following thing . If | |
| 02:54 | the variable A happens to be positive , which means | |
| 02:57 | it's bigger than zero , then that means that the | |
| 03:00 | Parabola opens up . And by the way , I | |
| 03:08 | have a computer demo right at the end of the | |
| 03:10 | lesson here that I'm gonna show you . That's gonna | |
| 03:11 | show graphically how how these parables behave . So stick | |
| 03:15 | with me to the end and you'll see it graphically | |
| 03:16 | interactive there . So if A . Is greater than | |
| 03:18 | zero , the problem opens up . I'll explain why | |
| 03:20 | in a second if A . Is less than zero | |
| 03:23 | , which means it's a negative number , then that | |
| 03:25 | means the Parabola opens doubt , which means it's a | |
| 03:33 | frowny face kind of problem . Now let me first | |
| 03:37 | give you a couple of examples . Let me get | |
| 03:39 | a couple of things down here , I'm gonna show | |
| 03:40 | you what this actually means , and then I'm going | |
| 03:42 | to show you why it works . And then I'm | |
| 03:44 | gonna do the computer demo so that you can see | |
| 03:46 | even more clearly what's going on here . So let's | |
| 03:48 | get some uh let's get some more information on the | |
| 03:52 | board before I can draw anything . One more thing | |
| 03:53 | I want to say , this is actually just as | |
| 03:55 | important as everything else here as the value of A | |
| 03:59 | increases whether it's positive or negative as it gets bigger | |
| 04:02 | and bigger and bigger . That's what this means here | |
| 04:04 | . This notation means as A gets bigger and bigger | |
| 04:06 | and bigger , then that means the parabola gets more | |
| 04:14 | narrow , gets more narrow . Now let me show | |
| 04:18 | you what I mean . I love talking of course | |
| 04:20 | , but I really like showing more than anything else | |
| 04:22 | . So let's do an example . Let's start right | |
| 04:25 | here . And let's do that . Basic problem here | |
| 04:29 | . I'm gonna draw a little sketch little sketch of | |
| 04:31 | a basic problem this Parabola is gonna be why is | |
| 04:35 | equal to X square . This parable is gonna be | |
| 04:37 | that basic beautiful parable that goes and touches and goes | |
| 04:40 | like this and so on . It opens upward because | |
| 04:44 | there's an invisible one in front of here , which | |
| 04:46 | means a . Is bigger than zero . The parabola | |
| 04:48 | opens up , right and I'm gonna show you a | |
| 04:51 | little more clearly why that's the case in a second | |
| 04:53 | . Right now , let's change it very slightly and | |
| 04:56 | let's take a a slightly different equation and we'll draw | |
| 05:01 | it on another access right here . Yeah , let's | |
| 05:04 | say that y is equal to two X square . | |
| 05:08 | So you see what we've done is we've replaced a | |
| 05:10 | instead of with one here , we've made it bigger | |
| 05:12 | . So what do you think is gonna happen ? | |
| 05:14 | Right as I put the numbers in front and you | |
| 05:16 | can kind of see what's gonna happen over here . | |
| 05:18 | The basic equation has a problem with the table of | |
| 05:21 | values like this . This was why is equal to | |
| 05:24 | X squared . If I make it , why is | |
| 05:26 | equal to two times X squared , then what's gonna | |
| 05:28 | happen is I'm gonna get all of these values that | |
| 05:30 | I'm gonna put the value of X in . I'm | |
| 05:32 | going to calculate the answer and whatever I get and | |
| 05:34 | then I'll have to multiply it by two . So | |
| 05:36 | every one of these things is gonna be multiplied by | |
| 05:39 | two , which means they're all gonna be bigger at | |
| 05:41 | the same value of X . All of these points | |
| 05:43 | are going to get bigger , which means they're gonna | |
| 05:45 | be shifted up . Which means when you have a | |
| 05:47 | bigger number here , the parabola is gonna get steeper | |
| 05:50 | like this , it's going to close up a little | |
| 05:52 | bit , something like this . Right ? And now | |
| 05:56 | you can see why I know a lot of books | |
| 05:58 | tell you well when the number gets bigger it gets | |
| 06:00 | bigger . But you don't often think about why the | |
| 06:02 | reason is because when compared to the basic Parabola , | |
| 06:05 | if you have a two in front and every number | |
| 06:07 | you get out of this thing is multiplied by two | |
| 06:09 | . Which means instead of nine it's gonna be 18 | |
| 06:12 | instead of four . It's gonna be eight here instead | |
| 06:14 | of one , it'll be two and so on and | |
| 06:16 | so on . So every one of these points will | |
| 06:17 | be shifted up . Which means the thing will be | |
| 06:20 | steeper . All right , let's take a look at | |
| 06:23 | another . Another guy here . Let's say that it | |
| 06:28 | is . What did I choose here ? Six X | |
| 06:32 | squared . So not one X squared . Not two | |
| 06:34 | X squared , but six X squared . That means | |
| 06:36 | that every point that comes out of the X squared | |
| 06:38 | part here , everyone in that table gets multiplied by | |
| 06:41 | six . So that means that they all get shifted | |
| 06:43 | up . And that means that this problem is going | |
| 06:45 | to be incredibly steep , something like this . I | |
| 06:48 | don't know exactly , I haven't drafted but it's gonna | |
| 06:49 | be much steeper than this one . In fact it's | |
| 06:51 | probably gonna be even more narrow than the way I've | |
| 06:52 | drawn it here because it's basically six times as steep | |
| 06:56 | as this one like this . All right now it | |
| 06:59 | goes the other way . So this is one X | |
| 07:01 | square two X squared 66 square notice I left a | |
| 07:04 | little space here in front because I want to draw | |
| 07:06 | one more . But I wanted to I wanted to | |
| 07:08 | do it at the end . What if I did | |
| 07:11 | the following equation ? Mhm . What if I did | |
| 07:15 | Why is equal to one half X . Squared ? | |
| 07:18 | So you see one X squared two X squared 66 | |
| 07:21 | square but this is a half X squared . Which | |
| 07:23 | means that if I were to take the table of | |
| 07:25 | values and if I put one half in front , | |
| 07:27 | then whenever I get out like the nine would be | |
| 07:29 | cut in half . So that would be 4.5 , | |
| 07:32 | this would be cut in half , that would be | |
| 07:33 | it to , this would be cut in half , | |
| 07:35 | which is a half and so on . So it | |
| 07:37 | means all of these points instead of being scrunched up | |
| 07:40 | , they're going to actually be coming down and kind | |
| 07:42 | of flattening out . So in this case instead of | |
| 07:45 | getting narrower this way because it's one half here , | |
| 07:47 | this parable , it gets much more lazy and kind | |
| 07:51 | of like opens up more broadly because all the points | |
| 07:53 | that we're here get pushed down . So the thing | |
| 07:55 | opens up like a flower kind of so as a | |
| 07:59 | gets larger , the parabola gets more and more narrow | |
| 08:02 | and as a is larger than zero , which is | |
| 08:04 | all of these cases , all of these are bigger | |
| 08:06 | than zero . Then the thing opens up and you | |
| 08:09 | can see why because if A is bigger than zero | |
| 08:11 | , then these numbers are all still positive . So | |
| 08:14 | everything is still going to open up as it does | |
| 08:16 | . It just changes the shape of the graph . | |
| 08:18 | Right ? Important for you to remember , because as | |
| 08:20 | we get into more complicated discussions , I don't want | |
| 08:23 | you to have to think , oh , a is | |
| 08:24 | bigger , what's gonna happen ? I want you to | |
| 08:25 | have an intuitive understanding when a is positive , it's | |
| 08:28 | a standard parabola . But as it gets bigger and | |
| 08:30 | bigger and bigger , it closes up because it's getting | |
| 08:33 | steeper and I'm trying to explain that here . Now | |
| 08:35 | let's take a look at the other case . What | |
| 08:37 | happens if A is less than zero ? We say | |
| 08:39 | that the parabola opens down . so let's do that | |
| 08:43 | one real quick and try to explain why . Alright | |
| 08:47 | , and then we'll do our computer demo . So | |
| 08:49 | here we have the equation and I'm gonna draw it | |
| 08:52 | right down below and this one's gonna be not why | |
| 08:55 | is equal to X squared ? It's gonna be y | |
| 08:57 | . Is equal to negative X . Square . So | |
| 08:59 | what have I done here here ? A the value | |
| 09:02 | in front of the export is a positive one here | |
| 09:04 | , the value in front of the export is a | |
| 09:06 | negative one . So it's the same absolute value positive | |
| 09:11 | one , negative one , but it's just negative instead | |
| 09:13 | of positive . So it should have the exact same | |
| 09:15 | shape , It should open up the same way , | |
| 09:17 | but it's going to open up down so it's going | |
| 09:20 | to look something like this and I can't draw things | |
| 09:23 | perfectly , but it's going to go up to a | |
| 09:25 | maximum value here and then down now I want to | |
| 09:27 | explore why with you . Why does it open down | |
| 09:30 | ? Why does it open down when this is negative | |
| 09:31 | here ? Because let's go back to our basic problem | |
| 09:34 | . Everything comes back to the basic problem . These | |
| 09:36 | are the table of values for the basic problem instead | |
| 09:39 | of why is equal to X squared . If I | |
| 09:41 | made it , why is equal to negative X . | |
| 09:43 | Squared ? Then what would happen is for everything that | |
| 09:46 | comes out of the X squared . If I stick | |
| 09:48 | this in I get a nine stick this and get | |
| 09:49 | a four and so on . I'm going to get | |
| 09:51 | that out but then I'm going to multiply by negative | |
| 09:53 | one . So that means that this output would not | |
| 09:55 | be nine , it would be negative nine . This | |
| 09:58 | one would be negative for this would be negative one | |
| 10:00 | , you can't have a negative zero , so it's | |
| 10:02 | gonna be zero and then again this one is negative | |
| 10:04 | one negative four negative nine . So what happens is | |
| 10:07 | you take all of the positive values that you had | |
| 10:09 | for the positive version of the problem and you stick | |
| 10:12 | a negative sign on there , which means you take | |
| 10:15 | all of these points and you map them down below | |
| 10:17 | this positive nine becomes a negative line . This positive | |
| 10:20 | war becomes a negative for this positive one becomes a | |
| 10:23 | negative one . Same thing happens on this side . | |
| 10:24 | So then the parabola opens up downward . That is | |
| 10:27 | why the parabola opens downward when the coefficient in front | |
| 10:31 | is negative . It's because you're taking the basic parable | |
| 10:33 | of the X . Squared and you're sticking negative signs | |
| 10:36 | right on the outside of it . And those are | |
| 10:38 | the points that you have to plot . That's why | |
| 10:39 | it opens up downward . Alright . It's something that | |
| 10:42 | isn't exactly taught in every book or every class . | |
| 10:45 | It's kind of like they tell you to memorize , | |
| 10:47 | hey this thing opens down but they never really tell | |
| 10:49 | you why . And so that's why I'm trying to | |
| 10:50 | do here now . What do you think is gonna | |
| 10:51 | happen if I say not two X squared negative two | |
| 10:55 | expert . Well the negative sign means it's going to | |
| 10:59 | open down but the value of the coefficient , the | |
| 11:02 | absolute value of it is a two , which means | |
| 11:04 | it's gonna be steeper which means it's gonna be a | |
| 11:06 | mirror image of what I have up above here . | |
| 11:08 | So it's gonna be steeper than this one or I | |
| 11:10 | should say crunched up a little bit more , something | |
| 11:14 | like that . So I'm trying to draw a mirror | |
| 11:15 | image of what I have here , the negative sign | |
| 11:18 | reflects it downward and the two makes the value steeper | |
| 11:22 | as we've discussed before and then the last one we're | |
| 11:25 | gonna do . You can totally guess what's going to | |
| 11:26 | happen here . Not a big surprise if you have | |
| 11:30 | instead of six X squared uh negative six run out | |
| 11:34 | of space . Let me go and fix that one | |
| 11:36 | , quit . Uh negative six X squared . What's | |
| 11:41 | that going to look like ? It's gonna look exactly | |
| 11:43 | like this but flipped upside down so I'll try my | |
| 11:45 | best , not gonna probably do a great job , | |
| 11:46 | but basically the problem will be even steeper than these | |
| 11:49 | but mapped downward . So there shouldn't be any confusion | |
| 11:52 | up to this point . I've tried to outline it | |
| 11:54 | to make it as clear as I can . Parabolas | |
| 11:57 | are in general a times X squared . This is | |
| 12:00 | still not the most general form , but it's more | |
| 12:02 | general than what we had before . It's centered in | |
| 12:05 | the origin . That hasn't changed . But the value | |
| 12:08 | of this coefficient in front changes how the thing looks | |
| 12:11 | . If it's a positive value , it always opens | |
| 12:13 | upward no matter what . If it's a negative value | |
| 12:15 | , it always opens downward no matter what . And | |
| 12:18 | the size of a as it gets larger and larger | |
| 12:21 | than probably gets more narrow , even if it's negative | |
| 12:24 | , if it's a bigger negative value , it just | |
| 12:26 | gets more more narrow in the negative way . Now | |
| 12:29 | , what I want to do is follow me over | |
| 12:31 | to the computer where I can show you a little | |
| 12:32 | more graphically and kind of interactively how this works . | |
| 12:35 | So follow me on right now . Hello , welcome | |
| 12:38 | back . So what we have is our computer demo | |
| 12:40 | , we have an equation F of X is equal | |
| 12:42 | to x square . This is our standard problem we've | |
| 12:44 | been talking about and this is the table of values | |
| 12:46 | that I have basically drawn on the board except now | |
| 12:49 | I'm going negative one , negative two , negative three | |
| 12:52 | . And also I include negative four and negative five | |
| 12:54 | . So I have a more more points here . | |
| 12:56 | But you can see that 149 Those are the same | |
| 12:59 | ones we had on the board . But now we | |
| 13:01 | have 16 and 25 . I have a little more | |
| 13:03 | points . Now . What if I change this curve | |
| 13:07 | ? So instead of X squared , it's two X | |
| 13:09 | squared . You see what happened is that probably got | |
| 13:11 | steeper . And the reason I got steeper is because | |
| 13:13 | all of these numbers got bigger , they got multiplied | |
| 13:16 | by two . If I go back to this one | |
| 13:18 | , you can see the 25 was right here , | |
| 13:20 | negative five comma 25 . And then when I multiply | |
| 13:23 | it by two , it becomes negative five comma 50 | |
| 13:26 | . And you can see all of these numbers because | |
| 13:28 | this was a 99 times two is 18 for instance | |
| 13:30 | , they all get multiplied by two . Uh No | |
| 13:32 | matter what . And so as I go past that | |
| 13:34 | this is times three and so on , I can | |
| 13:36 | crank this thing up and the problem just becomes more | |
| 13:38 | and more narrow . Now it's 10 times uh every | |
| 13:42 | point is multiplied by 10 . So it was 25 | |
| 13:45 | . Now it's 250 up here for this value and | |
| 13:48 | so on and so on . And that is why | |
| 13:49 | the probably gets more and more steep there . And | |
| 13:52 | the table of values kind of shows that now let's | |
| 13:54 | go back to zero . What do you think is | |
| 13:56 | gonna happen when we go to the negative direction ? | |
| 13:57 | Well here , first of all , we have to | |
| 13:59 | go through 00 times X squared means that basically you | |
| 14:03 | have no values because the value of zero , no | |
| 14:06 | matter what . So this is the flat horizontal line | |
| 14:08 | . It's not a problem at all . Which is | |
| 14:10 | what we said uh in the lecture there . But | |
| 14:12 | as we go negative negative X squared means you now | |
| 14:15 | have a frowny face Parabola upside down . So all | |
| 14:18 | of those points for the positive proble simply have a | |
| 14:21 | negative sign on them . And that is why it | |
| 14:23 | maps like this to prove that to yourself . Let's | |
| 14:25 | go through . You can see you have 149 16 | |
| 14:28 | 25 . And then here you have uh negative one | |
| 14:32 | negative four negative nine negative 16 negative 25 for both | |
| 14:35 | cases . That's why it goes negative like this . | |
| 14:37 | And then you can Make it steeper and steeper in | |
| 14:40 | that direction . Again we can go to negative 10 | |
| 14:42 | . You can see you have your negative 250 here | |
| 14:44 | . So that is why parabolas or how probable is | |
| 14:47 | open and get steeper when they're positive , the larger | |
| 14:51 | the value in front makes it steeper . They open | |
| 14:53 | downward , were in their negative and as that value | |
| 14:55 | gets larger they get steeper in that direction as well | |
| 14:58 | . Now I'm gonna get a little bit ahead of | |
| 15:00 | myself , I'm going to talk about this a little | |
| 15:01 | more in the next few lessons , but since I'm | |
| 15:03 | here , I want to show you that these basic | |
| 15:05 | problems are always centered at the origin here . And | |
| 15:09 | so I'm just telling you that what's in front basically | |
| 15:11 | changes the steepness of the thing and if it opens | |
| 15:13 | up or opens down , but it doesn't really matter | |
| 15:16 | where the parable that lives . Let's go move the | |
| 15:18 | parabola over here . We haven't talked about this much | |
| 15:20 | yet , but I'm gonna move it over here . | |
| 15:22 | So here have X squared plus two , X plus | |
| 15:24 | two . It's still a parabola and it still opens | |
| 15:26 | up because the coefficient in front of the X squared | |
| 15:29 | here is a positive one . It still opens up | |
| 15:32 | as I increase the coefficient in front of the X | |
| 15:36 | squared term , this number right here , the five | |
| 15:38 | , it's getting bigger and bigger , which just basically | |
| 15:40 | changes that . The problem is getting steeper and steeper | |
| 15:43 | and steeper . Right ? And as I go in | |
| 15:45 | the negative direction , now it turns into a line | |
| 15:48 | there , but when I go negative , it turns | |
| 15:49 | into a problem over here making it an upside down | |
| 15:52 | parabola . And as I make that number bigger and | |
| 15:54 | bigger in the negative direction , it gets steeper and | |
| 15:56 | steeper and steeper . So here's your basic proble uh | |
| 16:00 | here being in the positive sense , opening up in | |
| 16:03 | the negative sense , opening down . So let me | |
| 16:04 | go and reset this guy uh to where it should | |
| 16:08 | be . And here's your basic parabola . Now , | |
| 16:11 | let's go back to the board and close the lesson | |
| 16:14 | . All right , welcome back . I like to | |
| 16:16 | draw pictures on the board , but I really think | |
| 16:18 | the computer demos make things so much easier to understand | |
| 16:20 | and a lot of cases . And that is the | |
| 16:22 | basic idea that as you make this number in front | |
| 16:25 | of the X squared term larger and larger , the | |
| 16:26 | thing gets steeper and steeper and it actually , as | |
| 16:29 | I showed you in the computer , doesn't even matter | |
| 16:31 | if the problem is located right at the origin , | |
| 16:33 | wherever the parabola is . The only thing that governs | |
| 16:36 | if the thing opens up or down is going to | |
| 16:39 | be if the coefficient in front of the X squared | |
| 16:41 | term is positive or negative . Also , the steepness | |
| 16:44 | of the parabola is governed by the steepness of this | |
| 16:47 | term . If you get the whatever is in front | |
| 16:49 | here , larger and larger and larger , the problem | |
| 16:51 | will close up like this the same thing if it's | |
| 16:54 | upside down . So I hope you've enjoyed this lesson | |
| 16:56 | . Make sure you understand every uh every topic here | |
| 16:59 | , sketch some of these yourself and then follow me | |
| 17:01 | on to the next lesson where we're gonna start shifting | |
| 17:03 | the parabola around . See we haven't really shifted it | |
| 17:06 | in the lesson here anywhere . We have just kept | |
| 17:08 | it at the origin and looked at what happens when | |
| 17:11 | it opens up and down . But now we want | |
| 17:12 | to start to talk about what happens when you shift | |
| 17:14 | the parabola in different parts of the Xy plane . |
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OVERVIEW:
05 - Graphing Parabolas - Opening Up and Down (Quadratic Equations) is a free educational video by Math and Science.
This page not only allows students and teachers view 05 - Graphing Parabolas - Opening Up and Down (Quadratic Equations) videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.
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