Up next in 10
Welcome back to our Algebra 2 class, in today's lecture we'll continue and conclude chapter 2.6 today and look at part of this section as we discuss how to write and graph absolute value functions. This can be a bit confusing for some students and I'm hoping that this video will be helpful to you.
Chapters:
00: 00 Introduction
00:45 How to express an absolute value function in its piecewise form
02:55 How To graph an absolute value function
Show More Show Less View Video Transcript
0:00
so uh we're going to move on from the
0:03
pie wise Define function to another type
0:06
of pie wise function now these are
0:08
called absolute value functions and they
0:12
are not that hard you just have to
0:14
understand how to build them okay so
0:18
quickly I will explain like absolute
0:20
value of x is always what positive right
0:24
because that's what we know now here's
0:26
the thing like when I have absolute
0:28
value of x as a function I to graph it
0:31
it's a complex process cuz I first have
0:33
to be able to express this
0:35
function in its uh different um part for
0:39
example I don't know the value of x do I
0:43
I do not know what x is because X is
0:45
just X so now the question is absolute
0:48
value of x will be equal to x if x is
0:51
positive do you understand what that
0:53
means we need to understand that because
0:56
if you don't understand that it's hard
0:57
to grasp the uh this kind do you know
1:00
why is X x is positive
1:05
why abute value it's be all right so
1:09
yeah so for example what is absolute
1:11
value of two two two right what's
1:15
absolute value of -2 two right so
1:17
basically absolute value of x is going
1:21
to be x if x is positive meaning if x is
1:24
a positive value it's going to be the
1:26
same thing correct M now it's going to
1:28
be zero if x is is zero that makes sense
1:31
right because I just have to plug it in
1:33
here correct if x is zero because zero
1:36
is my what again remember zero is my
1:39
base my base my limiting value that's
1:41
the everything is built around is zero
1:43
here right now here you Bex if x is
1:47
negative what am I trying to say
1:49
here um um it's the absolute Val because
1:54
the absolute value is always positive
1:56
exactly so if x is negative thetive X
1:58
would be positive that's right did yall
2:00
get that right so for example I put
2:03
value of2 that gives me two
2:06
right -2 is what is really -2 that's
2:10
what it is right so I'm saying here the
2:14
absolute value of x ISX if x is negative
2:17
if x is negative I have to add another
2:19
negative to the X to make it what a
2:22
positive value does that make sense
2:25
right so this is why I I created this uh
2:29
piecewise function here so absolute
2:31
value of x is X suppose X is positive
2:35
and it's going to be zero if x is z and
2:37
it's going to Bex if my x value is
2:40
negative because I don't know what x is
2:42
so I have to consider all three options
2:45
does that make sense right now this is
2:49
where uh some of us get like a little
2:51
bit like confused now we have to graph
2:53
this function correct now after value
2:55
function is a fun function because it
2:58
always look like a V or if V that's
3:01
reverse so I love this function so now
3:04
if I to build this function right how
3:06
many tables am I going to make three
3:08
three table thank you very much right so
3:12
I have X Y now remember let me explain
3:15
this
3:17
again this is not this x here
3:20
means in other word Y is equal to x y =
3:24
0 and Y = tox that's what this means
3:28
right because f X and
3:31
Y mean the same thing correct we talked
3:34
about that in our previous chapter okay
3:36
so now here what would be my first value
3:39
for x zero zero choose another one one
3:43
so when X is zero Y is and when X is one
3:47
y is one right so when X is zero Y is
3:51
zero right and when X is one y is one
3:54
now here's the problem so I'm supposed
3:56
to be going where in this direction do I
3:58
go past this no no right so now I will
4:01
have one here normally open circle right
4:03
but you going to see that we we going to
4:05
eliminate that open circle in a minute
4:07
now now when X is zero Y is what zero
4:11
zero right and when X is one y will be Z
4:16
zero X is z so 0 0 is just one we're not
4:19
going to change that so it's going to be
4:20
right
4:21
here right because when X is z y is what
4:25
zero we only have one option cuz X is z
4:28
y is z so that's that's only one correct
4:31
and now we are here so I'm going to
4:33
start again with what Z zero when X is z
4:36
y will
4:37
be which is zero and I choose another
4:41
one and then y will
4:44
be posi one y 0
4:46
01 1 and
4:49
look this is the absolute value function
4:52
again it's a beautiful function if you
4:54
know how to slice it correctly right
4:56
it's always going to look like a V or
4:59
it's going to be reverse it depends on
5:00
what we have okay now this example is
5:03
pretty simple but I'm going to do
5:04
another one and then we will understand
5:07
how to build this function once and for
5:08
all okay now coming here right so
5:15
how without even having this here let me
5:18
see if I can establish this right this
5:19
is what I have I'm starting with this
5:22
absolute value of x - 2 suppose this is
5:25
what I'm I'm giving you I say graph this
5:27
function right the first step is to
5:30
build our
5:33
piecewise component correct so absolute
5:37
value is x -2 so we're going to go like
5:39
this just like the other
5:41
one ABS value of x will be what x - 2 if
5:46
what somebody help me here if x is
5:50
greater than Z no x - 2 is greater than
5:53
0 right x -2 because it's whole thing
5:57
it's not just X if this is positive the
6:00
abute Valu is positive right if x - 2 is
6:04
more than Z does that make
6:07
sense right and you'll be what you'll be
6:11
zero if x is equal to
6:16
what
6:18
no posi two you mean because if I set
6:22
this equal to zero I add two it would be
6:25
two right x = 2 that make sense yes or
6:30
no no all right let's build it again
6:32
let's confused all right let me explain
6:36
right the point not Z so here right this
6:40
is what we have
6:41
right absolute value of this number will
6:43
be what x - 2 if this number is
6:47
whater z greater than zero right there
6:50
you go in other words if this number is
6:52
greater than what zero two two two right
6:56
it's the same thing I'm saying the same
6:58
thing here does that make sense
7:00
right now so let's say this is our first
7:03
option right
7:05
now what if this thing is zero the
7:09
inside is zero right x - 2 is 0 that
7:12
mean X is equal to what zero zero X not
7:16
not z x equal to pos2 for this to be
7:20
equal to zero that mean X has to be
7:22
equal to what two two right so therefore
7:27
this will be zero if x is is equal to
7:30
two does that make sense yes or no not
7:33
really just keep on going we'll figure
7:35
it
7:38
no I want you to show I want to show you
7:40
what's actually going on here right
7:42
confus is the is the point not on the y
7:45
axis then it's on the x axis uh which
7:49
point um the um um the the point of the
7:54
absolute value function is on the x-axis
7:57
yeah yeah so here so what you need to do
8:00
is this this number
8:03
here is what is
8:05
causing this absolute value to change
8:07
right so now I want absolute value is
8:10
always going to be positive correct
8:12
right so now if this number inside is
8:15
positive then this is going to be the
8:18
same right so absolute value of f
8:21
ofx is going to be x - 2 if x -2 is
8:27
positive yes or no
8:30
yeah does that make sense yeah okay
8:32
because if this is positive then you be
8:34
this one right in other
8:37
words x is more than two because I'm
8:40
just adding two on both Sid does that
8:41
make sense to this point yes right all
8:44
right now what if this number inside was
8:49
Zero what if this was
8:52
zero right that
8:55
mean X is equal to
8:58
what two because you going to do x - 2 =
9:02
2 0 x =
9:05
2 right yeah if this is zero that mean X
9:09
is equal to 2 correct now wait would it
9:13
be
9:14
-2 x - 2 is equal to0 you add two on
9:17
both side that gives you x =
9:21
2 if this is equal to
9:24
zero yeah solve for
9:27
x w what would it
9:30
two does that make sense now to anybody
9:33
yes or no all right now what if this
9:35
number was negative right what if this
9:38
is like
9:40
this that means X is what x - 2 is less
9:44
than Z right which another word X is
9:48
what less than two that's all this mean
9:51
so this is how you build this function
9:54
is x - 2 if the number is positive is 0o
9:57
if x is 2 and therefore is
10:00
the opposite of this number if x is less
10:02
than two does that make sense now all
10:06
right now we can build our
10:12
graphs right XY XY X Y now here I'm
10:17
going to start X is what my first value
10:20
will
10:21
be yes I'm waiting two two right and X
10:27
is two Y is what
10:30
zero zero and another value when X is
10:33
three Y is 3 - 2 1 right here we have it
10:38
when X is zero when uh X is 2 Y is what
10:42
Z zero right and here we're going to do
10:44
the same thing when X is we're going to
10:46
start at two Y is what Z zero and when X
10:50
is uh take another one less than two One
10:53
X Y is 1 - 2 is 1 two negatives that's
10:57
positive right and then we go back here
11:00
we do the same
11:02
thing the rest is just understanding
11:05
this process so when X is
11:07
two Y is zero right and when X is 3 Y is
11:13
1 right here right so and I want X to be
11:17
more than two so I'm going to go this
11:19
direction right and then when X is 2 Y
11:22
is z I already have that over here and
11:25
then when X is one y is One X X is one y
11:30
is one I'm
11:33
here this is the function it's always
11:36
going to look like a d right if your
11:38
function does not look like a V guess
11:40
what something here right okay it's
11:44
going to look like this so the the most
11:46
important part is knowing how to build
11:49
the function based on the number that's
11:51
given so we're going to do several
11:52
examples and then tomorrow we're going
11:54
to do some more examples I'm going to
11:55
stop here and I'm going to let this you
11:57
can marinate on that right I don't want
11:59
to move on until we all understand this
12:01
so let's let's do that let's work some
12:03
more examples and then we're going to go
12:05
from there
#Teaching & Classroom Resources
#Mathematics
#Other