05 - Graphing Parabolas - Opening Up and Down (Quadratic Equations) - Free Educational videos for Students in K-12

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 .

Summarizer

DESCRIPTION:

Quality Math And Science Videos that feature step-by-step example problems!

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.