Up next in 10
Parents Functions And Transformations Part 1: Translations
22 views
Feb 19, 2025
We've learned how to analyze and used relations and functions. In Section 2.7, which we're currently in, we'll learn how to identify and use parent functions and describe transformations of functions. Chapters: 00:00 Introduction 00:43 What is a parent function? 01:19 What are the main parent functions? 02:59 What is a translation? 03:38 Using translations on constant functions 05:39 how to use translations on linear functions 08:48 how to use translations on absolute value functions 14:16 how to use translations on quadratic functions
View Video Transcript
0:00
I'm praying you at least you can pause
0:01
for a little bit all right next time
0:03
trying to do that please what next time
0:05
when I'm praying pause whatever you're
0:07
doing cuz prayer time we are saying we
0:09
are kind of giving time to God right I'm
0:11
talking about God yeah I know but I'm
0:14
praying now so you have to like at least
0:16
give me that one minute right am I right
0:21
or am I wrong on that uh yeah I talk to
0:24
you whatever you want to say but I'm
0:26
just saying it would be a good thing to
0:27
do anyway we're talking about parent
0:30
functions and
0:32
transformation right parent functions
0:34
and transformation so now the first term
0:37
that we we are encountering today is the
0:40
parent function what does that mean
0:43
parent function right is pretty much
0:47
what it means it's the daddy right of
0:49
the other functions that are going to
0:50
come after that it's the daddy function
0:53
right the D function so in mathematical
0:56
term is the simplest of the function in
0:59
the family
1:00
and as a graph it is the simplest of the
1:02
graphs in the family because everybody
1:05
comes from somebody right but God is the
1:09
ultimate Giver right he's the one that
1:11
started us all but again we have like in
1:15
this chapter we're going to talk about
1:17
four main function four main parent
1:20
functions right we're going to talk
1:22
about the constant function we're going
1:24
to talk about the identity function
1:27
we're also going to talk about the
1:29
absolute Val value function your
1:31
favorite and the quadratic function
1:33
which is another favorite right so we're
1:36
going to talk about how this function
1:38
pretty much uh play anal role in any
1:42
other function that's going to be built
1:44
based on who they are right for
1:49
example I come from a dad right so he is
1:54
pretty much if I was to be like very
1:56
like direct here's the reason why I am
1:59
right so these functions are the reason
2:02
why any other function that is
2:04
built is yes sir what do you mean you
2:06
came from your dad like if it wasn't for
2:10
my dad I wouldn't be here what about
2:12
your mom yeah her and my dad if it was
2:16
for them I wouldn't be right it takes a
2:18
man and woman to produce a child right
2:21
it takes his function to produce another
2:23
function now unfortunately the functions
2:25
are inanimate they can't produce like
2:28
that we have to produce stuff by
2:30
Building Things upon them right all
2:34
right so what's funny I'm just
2:38
serious
2:42
serious
2:43
right so now if you look here CU Cole
2:47
asked a very particular question that
2:49
required me all my brain man to come up
2:51
with solution and answer that would not
2:53
be compromising to my teaching anyway so
2:57
translation is the first one we going to
2:59
talk about several Transformations right
3:01
a transformation is if you begin with a
3:04
function here how can you transform that
3:07
function to get something else right we
3:09
use several transformation we use
3:11
translations we use dilation we use
3:13
reflection and we use Symmetry and all
3:16
that kind of symmetries but here the
3:19
first one that we're going to discuss is
3:20
the translation so here's the thing if I
3:25
give you this function this constant
3:27
function right two right and and I'm
3:30
just going to show you a pretty basic
3:31
transformation if I say graph for
3:35
me
3:37
um
3:38
y = 4 what are you going to do move that
3:43
line up two two times right you move
3:45
that line of twice so basically what are
3:47
you doing you adding what to this two
3:50
units right two units because you
3:52
starting at what what's your starting
3:54
point two so you moving this guy one two
3:57
and then you just getting this here can
4:00
you explain it again yeah where did you
4:02
get the two from no no I'll just give
4:04
you an example if I ask you for example
4:06
to graph y =
4:08
4 based on this guy here based on this
4:11
as your starting function how would you
4:13
graph it you're going to add two unit
4:16
right and move up how many times two
4:18
unit up because you're going four up
4:21
right now let me twist it up a little
4:23
bit what if I say I want you to graph y
4:25
= z what would you do move it down move
4:28
it down have any unit two two unit if
4:32
you're here I'm starting at two again if
4:35
I'm here and I want to go y = z what am
4:38
I going to do subtract two subtract two
4:40
I'm going down two units so I'm going
4:41
down here one and two right so this is
4:43
called a
4:45
translation right translation I had a
4:47
teacher that used trying to be funny us
4:48
to tell me translate yourself five unit
4:51
here and four unit here meaning get out
4:53
of my room that's what he to tell us
4:55
right so he he used to like be trying to
4:57
be fine like that so if I say I'm here
4:59
if I translate yourself five unit up
5:02
what am I going to do 1 2 3 4 boom right
5:08
and I come out 1 2 3 four and five right
5:12
so translate I can translate to the left
5:15
to the right back and forward so the
5:19
translation is only those ways you can
5:20
go up down left and right so that's the
5:25
translation this is what the translation
5:28
is you don't move up you move down you
5:30
move left or you move right based on the
5:33
base function now what if I say to to
5:36
you guys graph y = x + 2 how would you
5:40
do that
5:42
guys x+2 your X move what how would you
5:46
move this using the translation how
5:47
would you move this graph yeah um two
5:50
unit up two unit up right does that make
5:52
sense so if I'm here I'm going to find
5:55
some key point do I need to do
5:56
everything no I'm going to find maybe
5:58
just that and that right I'm going to go
6:01
what one I know we were on that and two
6:04
I thought we on the top no we are here
6:06
okay right X plus two that's was
6:08
confused and then I'm going up and up
6:11
and then what I'm going to draw a line
6:13
here you see what I'm doing so all you
6:15
doing is Shifting this two unit up does
6:18
that make sense did everybody get that I
6:21
want to make sure we didn't get it okay
6:24
so this is called a translation do you
6:26
know what a translation means first
6:28
that's the word you know what
6:29
translation mean figure
6:32
down down left right right so now I have
6:35
a base you know what the base function
6:38
is Right parent function that's my this
6:41
I'm suppose I'm the parent function
6:43
right and now I want to bring forth a
6:46
second function right and I say okay T
6:49
your X now I want you to go x + 2 so
6:53
it's going to be one and what two and
6:55
I'm here that's what I'm doing this is
6:58
the parent I want you to do x + 2
7:01
meaning starting with the base function
7:03
which is X right move it two units up
7:07
because I'm adding okay so that means I
7:10
need to move this entire function up two
7:12
unit but I don't have to do for every
7:14
single point is that Mak sense I want to
7:16
find some key point and I'm going to
7:18
move those point so I'm going to use for
7:20
example the middle this guy and this guy
7:22
I just need to move these up and then
7:24
draw the line so if I'm here if I'm
7:25
going two unit up I'm going to go one
7:27
two this one is going to end up here
7:30
correct one two this one is going to end
7:33
up here one two this one is going to end
7:35
up here and I draw a
7:37
line and this is x + 2 does that make
7:40
sense now right so this is how you move
7:43
the base function okay now you can start
7:45
with the same base function and go uh
7:49
let's say now I want you to do y = x - 2
7:52
now what would you
7:54
do move it down right for here you're
7:57
going to move this down how many unit
7:58
two two one two this one two one two
8:02
this is two one two and then draw the
8:05
line all right I L translations they are
8:09
the best one translations are easy to
8:10
figure out right so now let's let's do
8:14
this absolute value right we already
8:16
have the base function you guys are
8:17
going to hait me right all that stuff
8:19
that we did last like 3 days ago you
8:22
basically remember I ask graph like X
8:24
absolute value of x + 2 you could have
8:26
started with a base function and just
8:27
move it up and down that's me something
8:30
wanted to like me she said teaching P
8:33
why I said well
8:38
because no no we we we still have to do
8:41
it we still have to do it right so now
8:44
go
8:46
ahead
8:48
over huh not for nothing you're learning
8:52
that I suffer I don't
8:54
want I didn't get any of the homework
8:57
there was no point it is point because
8:59
you have to learn how to do because it's
9:01
still
9:03
important no you can't do it this way
9:05
right now you build on it right so that
9:09
what you to graph this absolute value of
9:10
x + 5 so how would you do
9:13
it you move what we get study guides I
9:17
gave it to you this morning oh yeah you
9:19
did so I'm going to move I'm going to
9:21
move is this test No 1 2 3 4 5 what
9:28
else here right 1 2 3 4 5 here 1 2 3 4 5
9:35
and what boom boom and it goes up this
9:38
way so that's how you doing it okay does
9:40
that make sense this looks like a kid on
9:42
Microsoft Paint or something yeah anyway
9:46
so this is how you use a translation
9:49
translation is up down left right okay
9:52
so far we only doing the up down we
9:54
haven't done left and right left and
9:56
right is going to be left and right now
9:58
we're going to do left and right right
9:59
it's going to be interesting right so
10:02
now can we do that after our test no I'm
10:06
doing it right
10:08
now all right
10:12
so now let's do this here so let's
10:16
assume that my starting function again
10:20
is y equal absolute value of x you you
10:24
already know how this looks right now
10:26
the key thing is this you have to know
10:27
how this function looks first if if you
10:29
don't know what it looks like you won't
10:31
be able to build any other function so
10:33
it's important to know what it looks
10:34
like so absolute value of x looks like
10:36
what if from zero it goes like this
10:39
right so now what I want you to graph is
10:42
this I want you to graph this
10:45
first and I also want you to graph this
10:47
for me now I will explain something here
10:51
what is the difference between the
10:52
two um is the value the other is not the
10:57
what the other is not right now let me
10:59
explain this so that it makes sense to
11:01
you when the number that you are using
11:04
for your transformation is outside the
11:06
absolute value your transformation
11:09
occurs on the Y AIS does that make
11:14
sense right but when the number that you
11:17
trying to use to make the transformation
11:19
is within the parenthesis the
11:21
transformation is
11:22
under X AIS now this is awkward because
11:26
you're going to see what happens when
11:27
you're subtracting here you go right but
11:30
when you're adding you go left okay so
11:33
let me do both of them and then it make
11:34
sense to
11:35
you so let's graft this one first how
11:38
would you do Patrick what would you do
11:42
here you choose points on the
11:45
mhm you go you go you go oh you go down
11:51
right one 2
11:54
3 one 2
11:56
three one and two and three right and
11:59
and then boom so this here will be
12:03
absolute value of x - 3 now how do we do
12:06
the next one again let's start with this
12:09
one right watch this so this is our
12:13
parent function so how do we do the x -
12:15
3 so what will most of you do go right
12:19
go would you go left or right yeah left
12:21
you say right you say left left who says
12:26
left left right it's right it's
12:29
interesting how this works when you're
12:31
subtracting right you're going right but
12:34
when you're adding you go left oppos
12:37
it's opposite stupid right it's not
12:40
stupid it's just it's not stupid stupid
12:43
never going to to do this ever it's I
12:45
don't like to get that is the person who
12:47
made this de no it's not watch
12:51
here right so look here we're going to
12:54
go to the right right when you're we
12:57
going to the right so this going to go
12:59
where one and two and three right this
13:04
is going to go and one and two and three
13:08
and this is going to go and one and two
13:10
and three and then guess what
13:12
boom right so this is how you graph the
13:17
translation on the left and on the right
13:20
simp okay I want to stick to
13:23
translations to this so that way you
13:24
guys can kind of absorb it and really
13:26
like soak in it and then like get it
13:28
before move on to like something else
13:30
right now let's do some more examples
13:34
does this make sense to everybody yes
13:36
yes or no all right so now let me take
13:39
another example and then we're going to
13:40
see how you guys are going to do
13:43
this so now what if I have y =
13:48
x² + 5 and I have y
13:53
= x -
13:57
5² now you see
14:10
right you sure all right let's see this
14:13
what's my parent function I need to
14:15
identify what's the parent function oh
14:17
wait you mean from the parent function I
14:19
was trying to get from function a to
14:20
function B no it's no from the parent
14:25
function all right first what's my
14:27
parent function here let's the par what
14:29
is it y y = what x² as everybody agree
14:34
that is y = x² right so now if I'm here
14:38
right this is y = X2 how would I graph
14:41
the first
14:43
one you yeah this is X2 + 5 so how would
14:48
I graph it I said no clue you know it
14:52
than this is the parent function I'm
14:55
doing plus five
14:59
you will that's it but why is it squ
15:03
it's because this is the parent function
15:05
you know what the parent function is
15:06
Right parent this is the what starting
15:10
point now from the start the square
15:12
matter the square doesn't matter it's x
15:14
square right and I'm adding five so I
15:17
already have x s all I'm doing is what
15:20
adding so it's squ when it's squ yeah
15:23
it's when it's curved it's a square
15:25
function this is y = X2 right this is
15:29
your parent function this is y = x² this
15:33
is standard this is universal no matter
15:35
where you go China Mexico wherever you
15:38
go it's y = x² Africa wherever you are y
15:41
= x² will always look this way it don't
15:44
change it does not
15:47
change right so now y = X2 + 5 Stephanie
15:52
what do I do to graph it you go up five
15:54
five right so this one's going to go up
15:56
one 2 3 4 five this one's going to go
16:00
one two 3 4 five and then same here and
16:03
then you have the curve right does that
16:06
make sense all right now the difference
16:09
come here now I have y = x - 5
16:16
squared right it's still the same
16:18
function the P function is still what x
16:21
but now what the five is what the
16:24
changing number is in the parenthesis
16:26
not 25 not 25 so what do I do
16:29
I move the curve
16:32
how to the to the right right how many
16:36
times five unit
16:40
right it doesn't matter is this function
16:42
that I changed wait so if it's squared
16:44
then you just like make it a circle
16:48
don't you just Square the five I'm
16:50
pretty sure it's like a
16:54
you wait a second look this is your
16:57
parent function
16:59
okay start with a parent function which
17:02
is the base function everything is going
17:04
to be built around that right so if I
17:09
add five to this function that means I'm
17:12
moving what up right if I subtract five
17:17
from this function it will look like
17:18
it's X