Transformations of Functions - Free Educational videos for Students in K-12

Transformations of Functions - By The Organic Chemistry Tutor

Transcript


00:00	in this video , we're going to talk about functions
00:02	, transformations and things like that . So let's say
00:08	if we have the function ffx , let's say we
00:14	add to to him , What type of shift do
00:19	we have here ? If you add 2 to it
00:22	? This is known as a vertical shift . The
00:27	graph is gonna move up two units . Likewise .
00:31	Let's say if you have X minus street , that's
00:33	a vertical shift down three units . Now , what
00:39	about F of X minus four . This is a
00:45	horizontal shift and it shifts Do you think it shifts
00:51	four units to the left or to the right ?
00:54	It turns out that this shift Is four units to
00:57	the right . If you set X -4 equal to
01:00	zero , X will equal four . So it doesn't
01:04	shift to the left . four units by shift to
01:05	the right . Now let's see if we have F
01:09	of X plus story , This would shift to the
01:12	left three units . If you set the inside equal
01:16	to zero , you'll get access equal to negative three
01:21	. No , there's some other ones that you need
01:22	to know as well . If you have a negative
01:26	on the outside , it reflects over the X axis
01:31	. Now if you have a negative on the inside
01:35	it reflects over the Y axis . And if you
01:40	have a negative on the outside and on the inside
01:43	it reflects over the origin . Now let's see if
01:48	we put it to in front of F of X
01:51	. This is known as a vertical mhm stretch .
01:57	And let's see if we put a fraction in front
01:59	of FX , it's going to shrink vertically . Now
02:06	if we put the two on the inside it's going
02:10	to be a horizontal . Not a stretch but a
02:13	horizontal shrink . So be careful of that one .
02:20	And if we put one half on the inside ,
02:24	this is going to be a horizontal stretch . So
02:32	those are some things that you want to keep in
02:34	mind in terms of transformations . So let's go over
02:37	some examples , let's say if we have the parent
02:40	function F of X is equal to X squared .
02:45	And basically , that's a problem that opens upward like
02:48	this . Now let's say if we wish to graph
02:54	this function X squared plus street , this is going
02:59	to be A vertical shift up three units . So
03:05	the graph is going to start at 03 and it's
03:07	going to look the same . Now , for example
03:12	, let's say if we wanted a graph X squared
03:16	minus two . Yeah , we're gonna have the same
03:19	type of graph but It's going to shift down two
03:22	units . And so it's going to look like that
03:27	. Now , what about this one ? Let's say
03:28	that Y is equal to the absolute value of X
03:31	. Mhm . So here's the parent function . Yeah
03:39	. Now what if we put let's say X plus
03:43	two , how is the graph going to look like
03:45	? Now in this case is going to shift to
03:49	units to left so it's gonna look like this .
03:54	So this point was at zero and now at this
03:57	point is that negative too ? Now what about this
04:02	one ? The absolute value of X . Spine stream
04:07	. So we're gonna have the same type of function
04:08	but it's going to shift three units to the right
04:12	so it's gonna look like that . Yeah . Now
04:19	what about this one ? Let's say that . Why
04:20	is equal to the square root of X . And
04:24	the parent function looks like this ? Yeah , go
04:29	ahead . And graph Y is equal to negative square
04:33	root X . And then square root of negative acts
04:38	. You can put it here and then we'll do
04:40	one more . So this one reflects over the X
04:46	axis . Therefore it's gonna look like this . This
04:50	one reflects over the Y axis and here's the y
04:52	axis . And so it's going to go that way
04:59	. And if we have a negative on the inside
05:01	and the outside it's going to reflect over the origin
05:06	. And so it's going to go towards quadrant dream
05:11	. Now a good way to like remember which direction
05:14	goes is to look at the science . We have
05:16	positive X . And positive wife Access positive in quadrant
05:20	one and four towards the right . And why is
05:24	positive in one and 2 ? So when X is
05:28	positive needs to go to the right and why is
05:30	positive if you go up ? So this is going
05:32	to go towards quadrant one . Yeah . Now for
05:38	the next one X . is positive wise negative .
05:40	So we're gonna go to the right and then why
05:43	is negative as you go down ? So that takes
05:46	us to quadrant four for this one , X .
05:49	Is negative but Y . Is positive , X .
05:52	Is negative on the left . Why is positive as
05:55	you go up ? So that's towards quadrant to And
05:59	for the last one X . Is negative and why
06:02	is negative ? So it's going to go towards a
06:05	quadrant for not quadrant for the quadrant dream , This
06:11	is quadrant four . This example now let's graph these
06:17	three functions using points the absolute value of X And
06:22	then two times the absolute value of X . And
06:26	then 1/2 . Mhm . So we're gonna have the
06:39	zero 1 12 , 2 -1 positive one negative to
06:45	positive too . And so that's the graph for the
06:49	absolute value of X . Now for two times the
06:53	absolute value of X it's going to be as follows
07:02	, We're still gonna have a zero . But when
07:04	X . Is one , why it's going to be
07:05	too When X . is to Y . is four
07:09	and then it's symmetric about the Y . Axis ,
07:12	so the right side and the left side will look
07:14	the same . So here we have a vertical stretch
07:19	notice that the Y . Values were doubled . And
07:22	so that's the effect of putting a two in front
07:24	of the function . For the last example we're going
07:28	to have a vertical shrink . Yeah , so at
07:46	one is going to be a half , At two
07:49	is going to be one . And so this is
07:56	a vertical shrink . The Y values were cut in
07:59	half , they're half of what they were compared to
08:03	that graph . So now you can visually see how
08:07	a vertical stretch appears and vertical shrink appears as well
08:12	. So for a vertical stretch , the why values
08:15	are increased for a vertical shrink , the Y values
08:20	were decreased . So now let's go back to this
08:24	graph , Y equals X squared . And I want
08:28	you to graph let's say actually let's use a different
08:38	example . Let's use the square root of X .
08:43	And let's use the square root of two x .
08:46	And also the square of one half X . Hopefully
08:51	I can fit all of them here . So I
08:53	only need the right side of the graph . Yeah
09:01	. So when X zero , Y zero when X
09:03	is one Lies one , the square root of voice
09:06	to . So in excess four wise too . So
09:09	that's the parent function . It looks something like that
09:24	. Now for this one when X zero , Y
09:26	zero when X is a half , Why it's going
09:31	to be one And when access to why is going
09:36	to be too . So notice that The X values
09:45	would decrease by two . Here it was four And
09:51	it had the same y . value of two .
09:53	But for the same white value , the X value
09:56	is now too . Now granted this graph will continue
09:59	to grow but I want to stop at the point
10:02	where the Y . Value is the same . So
10:12	I'm going to stop at this point here . So
10:16	you can see the effect that the graph has on
10:20	X . We said when X is a half ,
10:30	why will be one And when access to why is
10:33	4 ? So for the same y . value of
10:38	two , X . Is no longer for . But
10:41	it's too . So as you can see the x
10:44	value was reduced by two and that's why this is
10:47	known as a horizontal shrink , It shrinks the x
10:51	values by a factor of two . Now let's look
10:55	at the last example now When X0 , Y zero
11:08	. But when access to Why will be one and
11:13	when X is eight , half of eight is four
11:15	. The square root of force to So in excess
11:17	eight y will be tuned . So for the same
11:21	y value of two , X . has increased to
11:26	eight . So going from 4 to 8 . We
11:29	could see that X was increased by a factor of
11:31	two . So this is a horizontal stretch and you
11:35	have to make sure that the Y values are the
11:37	same . Otherwise the graph may look like a vertical
11:42	shrink . For instance , if I were to compare
11:48	this graph versus this graph , if you stretch it
11:51	out , this might appear as if it's a vertical
11:54	stretch compared to this one because this looks higher .
11:57	However , you need to compare the X . Values
12:00	for the same Y . Value . And if you
12:02	do that , you can clearly see that this is
12:05	a horizontal shrink and not a vertical stretch . And
12:12	this one you can see that it's a horizontal stretch
12:14	, not a vertical shrink . Now let's move on
12:18	. Let's work on some other examples . So make
12:21	sure you're aware of the parent functions . Let me
12:23	just run through them real quick . Mhm . So
12:28	this is the Graph four Y equals X . The
12:32	next one you need to be familiar with . As
12:36	we mentioned earlier in this video is Y equals X
12:40	squared . Yeah . Mhm . Yeah . And here's
12:45	another one . This is equal to this is why
12:49	equals X cube . Mhm . And you've seen this
12:53	one already , Y equals square X . Mhm .
13:04	Mhm . Mhm . Yeah . And then we have
13:10	the cube root of X . And the absolute value
13:13	of X . Now there's some other functions but these
13:21	are the main ones that we're going to go over
13:23	for now . So how would you graph this function
13:30	X -2 Squared Plus String . So , graphic using
13:35	transformations . So the parent function is x squared ,
13:40	which looks like this . However , we can see
13:43	that We have a horizontal shift two units to the
13:48	right And the vertical shift up three . So I'm
13:52	going to shift it to units to the right and
13:55	up three . Mhm . And then you can just
13:59	draw a rough sketch and that's it for that example
14:09	. Now what about this 1 ? Let's say if
14:13	we have three minus X plus two squares . So
14:22	once again we have a problem and this time I'm
14:27	going to plot it more accurately instead of using a
14:30	rough sketch . Now this is a vertical shift up
14:34	three . If you want to you can be right
14:37	it this way this is why is equal to negative
14:41	X plus two squared plus story . So we have
14:44	a vertical shift of three . Now we also have
14:48	a horizontal shift left to so the center or the
14:52	vertex of the problem , It's going to be a
14:55	negative to comment three which is here . Now there's
15:00	a problem open upward or downward . Normally it would
15:04	open upward . However , we do have a negative
15:08	sign , so it's going to open and downward ,
15:12	but let's get some points . Let's graph it accurately
15:17	. Now keep in mind the parent function is X
15:19	squared . So one square is one . That means
15:24	that if you travel one unit to the right from
15:27	your vertex , the next point will be down one
15:30	And one Unit 2 left . It's also going to
15:32	be done one . So if you plug in one
15:37	into X , the y value should be too I
15:44	mean that one but negative one because this is negative
15:48	one here . So for instance -1 Plus two is
15:56	1 . One squared is one , so 3 -1
15:59	is two . And that gives us at that point
16:07	Now two squared is equal to four . So as
16:11	we travel to units away from the vertex we need
16:13	to go down for . So the next point is
16:20	going to be 0 -1 and also -4 -1 .
16:26	Due to the symmetry the graph . And that should
16:29	be enough to get a decent sketch . Now ,
16:33	if you want to test it , let's use negative
16:35	for let's see if we get negative one . So
16:38	three minus negative four plus two squared . So negative
16:43	4-plus 2 . That's negative too . Negative two squared
16:47	is positive for three minus four is negative one .
16:50	So it does gives us that point . And if
16:53	you try zero you get the same thing . Zero
16:56	plus two is two , two squared is 43 minus
16:59	four is negative one . So this technique works if
17:03	you don't want to make a table and if you
17:05	want to draw an accurate sketch , Let's try this
17:10	one . Let's say that . Why is equal to
17:12	four minus the square root of three minus X .
17:20	Let's try an accurate sketch . Yeah , let's rewrite
17:28	it first . You need to see it like this
17:34	. So there's a vertical shift up four units and
17:41	there's a horizontal shift but it looks a little different
17:44	. Is it three units to the right Or three
17:47	units to the left . If you're ever unsure ,
17:51	set the inside equal to zero and soft rex ,
17:54	I'm going to take this term and move it to
17:56	the side . It's going to switch from negative X
17:59	. Two positive acts And so acts is equal to
18:03	positive three . So that indicates that We have the
18:07	horizontal shift to the right of three units . So
18:12	the starting point It's going to be at 3:04 .
18:17	Now the parent function is the square root of X
18:20	. And we have a negative on the outside and
18:23	a negative in front of the X . So will
18:26	the graph shift towards quadrant one towards quadrant to towards
18:32	quadrant three or towards quadrant for while X is negative
18:38	towards the left . And why is negative as you
18:41	go down ? So it's going to go towards quadrant
18:44	dream . So now that we know the direction in
18:49	which this graph is going to go keep my having
18:52	this negative sign here . It reflects over the X
18:54	. Axis and haven't here , it reflects over the
18:57	Y axis . So originally this is the square root
19:00	of X . If you reflect over the X axis
19:03	it looks like this and then if you reflecting over
19:06	the y axis it looks like that . So when
19:10	it's reflected both about the y axis , I mean
19:13	about the X axis and the y axis . It's
19:16	equivalent to reflecting about the origin . So we can
19:25	clearly see that it's going to go in this direction
19:27	. Now let's get some other points . So the
19:33	parent function is the square root of X . The
19:35	square root of one is one . That means that
19:39	actually move running it to left . We need to
19:41	go down one . So that's going to give us
19:43	the point 23 So if you plug in to you
19:48	should get a wide value of three . So 4
19:51	- the square root of 3 -2 , 3 -2
19:55	is one and the square root of one is 1
19:59	, 4 -1 Street . Which we do get .
20:07	Now the next best .2 uses for the square root
20:11	of four is too . So as we travel four
20:14	units to left , we need to go down soon
20:18	. So four units to the left will take us
20:20	to the x . value of -1 . And we
20:23	need to go down to so the Y value will
20:25	be too . And let's do it one more time
20:28	. The square root of nine is street . So
20:31	this is at three . So if we go nine
20:33	units to left , 3 -9 is -6 . That's
20:38	going to take us to this point and we need
20:41	to go down three . So we're starting at 44
20:43	ministries one . So we're gonna have the 0.-61 .
20:49	So if you plug in negative six , that should
20:52	give you one . So four minus square root three
20:55	minus negative six , three minus negative six is the
20:59	same as three plus six which is not . And
21:02	the Square Root of 9th Street and four Ministry is
21:05	one . So we do get this point , and
21:08	so now we can plot it . And so let
21:13	me see if I can do that a little better
21:16	. It should be something like that . My graph
21:18	is not perfect , but you get the point .
21:22	It has this general shape to it , and that's
21:28	how you can graph that particular function .

Summarizer

DESCRIPTION:

This precalculus video tutorial provides a basic introduction into transformations of functions. It explains how to identify the parent functions as well as vertical shifts, horizontal shifts, vertical stretching and shrinking, horizontal stretches and compressions, reflection about the x-axis, reflection about the y-axis, reflections about the origins and more. Parent functions include absolute value functions, quadratic functions, cubic functions, and radical functions. This video contains plenty of examples on graphing functions using transformations.

OVERVIEW:

Transformations of Functions is a free educational video by The Organic Chemistry Tutor.

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