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Welcome back to our Algebra 2 class, in our last session we learned how to model data using lines of regression. In section 2.6, we'll learn how to write and graph piecewise- defined functions, so buckle up and let's get started.
Chapters:
00:00 Introduction
00:45 What is a piecewise defined function?
01:50 How To graph a piecewise defined function
12:10 How To write a piecewise function given a graph
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0:00
discussing today so it's this is uh
0:03
quarter three I mean Quarter Two we're
0:05
going into so these functions are pretty
0:08
hard they're not easy to like detect
0:10
especially if you are like um you've
0:13
never seen this type of functions before
0:15
so it can get a little bit tricky but I
0:18
guarantee you if you really do what I
0:20
say then you won't have any problem with
0:21
these functions okay so they called the
0:24
first type of function we going to
0:26
discuss is called a pieace wise function
0:29
peie function okay so we also going to
0:32
talk about step functions and also going
0:35
to talk about absolute value function
0:37
but the first type is a pie wise
0:39
function so what is a pie wise function
0:42
so if you look at my board here what we
0:45
have is a coordinate system right we
0:48
have a coordinate system and then these
0:50
two functions here it's it's a single
0:52
function but that's broken down into
0:54
small parts okay now if you see this is
0:57
linear this is also linear but then
1:00
there is a gap in between them right so
1:03
by definition a p s function it is a
1:06
function that is written using two or
1:08
more Expressions okay and on the graph
1:11
of a piece function pie wise function uh
1:14
a do implies that the point here is
1:16
included right and when you have a
1:18
circle that mean that is not included
1:21
it's the same thing as your function
1:22
when you go X is less than five and when
1:25
you want to represent that function you
1:26
put a whole because the whole indicate
1:29
that the five is included and when you
1:31
say x is less than or equal to 5 you put
1:33
a dot indicate that dot the dot shows
1:36
you that uh the point is included it's
1:38
the same thing with piecewise functions
1:40
okay now we're going to learn how to
1:41
build them given the P wise function how
1:44
do you graph the function that's the
1:46
delicate part okay so I'll give you an
1:48
example here first example we have f of
1:52
x is given and FX is given and you see I
1:56
have this bracket here right so I have a
1:59
specific
2:01
uh domain for this function f ofx here
2:04
is equal to x - 2 if x is strictly less
2:09
than1 and is x + 3 if x is greater than
2:14
or equal
2:15
to1 why is this important this is going
2:18
to help us to draw this function okay so
2:20
what I'm going to do is I'm going to
2:21
delete this for now and I'm going to
2:23
show you how they got to that right so I
2:27
just wanted to show you this as an
2:28
example so to draw this function you
2:31
first start with a table of
2:34
values we going to make two table of
2:36
values and I I tell you why I did that I
2:39
want to separate these two
2:42
because each function has a specific
2:49
domain so f ofx is given right and we
2:54
are told that f of x is equal to x - 2
3:00
if x is less
3:01
than1 right again what type of function
3:04
is
3:05
this is it a linear function or not yeah
3:09
yes because it's written in the form
3:11
what MX plus b correct so because it's
3:14
written the form MX plus b is a linear
3:15
function yes
3:17
sir okay and to graph a linear function
3:21
all you need is two points you don't
3:22
need more than two points okay so now
3:25
what I'm going to do is this because
3:27
they told us that the function = x - 2
3:31
if x is less than1 I'm going to start
3:34
with1
3:36
right I know that this is not included
3:38
but I'm going to start with1 because
3:42
this is what they gave me as a domain
3:43
for this function and then I'm going to
3:46
choose another point a point that is
3:48
less than netive 1 I cannot choose
3:50
anything that's more than ne1 because if
3:51
I do that I'm going to mess up my
3:54
function does that make sense okay so I
3:57
have to choose a second value that is
3:59
less than 1 give you an example -2
4:03
-2 right and that's all you need really
4:06
to graph this function so now when xal
4:09
to1 you have to plug it into the
4:12
appropriate function you cannot plug
4:13
that in here okay so when X is1 I'm
4:17
going to plug it in I get -1 -
4:20
2 that is
4:22
-3 and when X is -2 -2 - 2 that is -4
4:27
right so now I'm when graphing that
4:30
that's what you have to be careful when
4:32
X is -1 Y is
4:34
-3 right so it's 1 2 3 right
4:42
here and I have to put a
4:44
[Music]
4:45
circle I have to put a circle why am I
4:47
putting a circle I need to use a
4:49
different
4:50
color when X is1 Y is3 why do you think
4:53
I'm putting a circle here CU it's just
4:55
less than because it's less than right
4:58
if it was less than or equal then I put
5:02
um if it was less than or equal I would
5:04
put a dot but because it's less than
5:07
I'll put a circle right and then when X
5:10
is equal to
5:12
-2 Y is equal to4 so I'm right here one
5:15
2 3 and four so right here right and now
5:20
I'm just going to draw my curve do you
5:22
see that right so I'm going down because
5:25
x equal to1 is right here I'm not going
5:29
above that because my domain is specific
5:32
is a specific domain is less than -1 so
5:36
I can't go beyond this point does that
5:38
make sense we have to be extremely
5:40
careful over there because X is1 my
5:42
domain is from 1 to Infinity so if I'm
5:45
going to graph this function I have to
5:47
be here I cannot go beyond this
5:50
point right
5:53
now this one is different so again I'm
5:56
going to start with negative 1 same
5:58
process when X is -1 I'm going to plug
6:02
it in here so I get1 + 3 that is
6:06
pos2 right and then I'm going to choose
6:09
another value that's more than1
6:11
example
6:12
Patrick two two right when X is 2 2 + 3
6:17
that is 5 so I only need two values
6:20
again this is a linear function I don't
6:23
need more than two values so when X is
6:25
-1 Y is 2 so one and two is right here
6:29
now your D because as you can see this
6:33
is more than or equal
6:34
to1 right it's not more than NE 1 is
6:38
more than or equal that mean negative 1
6:40
is included so I have to put a dot there
6:42
to indicate that does that make sense
6:45
now next when X is
6:47
2 Y is 5 1 2 3 4 5 right here right
6:52
again I'm going to go beyond this okay
6:56
because my domain tells me that if x is
6:59
greater than equal to1 the function is
7:01
equal to this so this is going all the
7:03
way to Infinity from1 to infinity and
7:05
then now this is the function so this is
7:08
how you you you draw what you call a p
7:11
wise function you have to be extremely
7:13
careful how you draw this because if you
7:16
choose the wrong value you're going to
7:18
get a graph that is not reflecting what
7:20
you given as a pie wise function all
7:23
right so you have to have the
7:25
appropriate uh unit so let's do an
7:28
example here no I'm not erasing it okay
7:31
I'm not erasing it before you can I just
7:32
wait can you just wait a sec
7:36
yeah yeah
7:40
okay can I raise this
7:42
stove
7:49
okay can yeah you
7:53
can that yeah all right
7:59
so now we have f
8:03
ofx is equal
8:07
to x +
8:10
2 if x is less than
8:14
zero and then f x is x if x is greater
8:19
than or equal to zero right that's
8:21
example number two so how we going to
8:22
graph graph this function we're going to
8:25
use the same process okay can I raise
8:28
this now yeah
8:32
okay so now we have another
8:43
function now remember when you draw this
8:47
function you always have to start with a
8:48
limiting point right my cut off value is
8:51
zero correct so code if I want to plug
8:53
in if I want to put the for the first
8:55
function we're going to call that
8:56
function one right if you want I choose
8:59
my my arbitrary values what will be my
9:02
first value here say it again you have
9:05
to start
9:07
at zero right you have to always start
9:10
at a value that's given to you because
9:12
that's the cut off all right it's almost
9:15
like that's your border if you start at
9:17
a value that's less than that or more
9:19
than that it's going to screw up your
9:21
graph okay always start with the
9:25
limiting value the cut off value okay
9:28
when X is Z what will be F ofx
9:32
shape zero you got to plug it in the
9:35
function so two two right that's
9:40
two and then I need another value that's
9:43
less than
9:46
zero1 right when X is1 I'm plugging it
9:49
in there 1 + 2 is 1 okay so now I have
9:53
my two values for this
9:57
function so I'm going to start here when
10:00
X is zero right Y is two right here so
10:04
what am I going to put there now a DOT
10:06
or a circle cires circle
10:09
right and then the reason why I'm
10:12
putting a circle is
10:14
why because it's less than zero Z is not
10:17
included okay if it's less than or equal
10:20
or more than or equal you put a dot but
10:23
if it's otherwise you put a circle okay
10:26
now when X is -1 Y is 1
10:30
right when X is1 Y is 1 I'm right here
10:33
so remember my domain is from what from
10:36
0 to negative Infinity so if I want to
10:40
draw my line it's going to go this way
10:43
and I will have this that make sense
10:46
okay and now I'm on to the second
10:49
graph and X is more than or equal to
10:52
zero so what value do I start with here
10:54
zero zero there you go zero so when X is
10:57
z what y
10:59
zero and then choose another one easiest
11:03
one it has to be more than or equal to
11:06
zero one right and then one so when X is
11:10
zero Y is zero and when X is one y is
11:13
one right here and then you go this way
11:16
so this is how the graph is going to
11:18
look so 0 0 1 1 and then make sure you
11:23
draw here because your domain is from
11:26
zero to positive Infinity right
11:29
so therefore we have to make sure we
11:31
putting the the dot here to indicate
11:33
that Zer is included all right so again
11:37
these are not really hard to do once you
11:39
understand the gist of it remember
11:41
always put
11:44
the the limiting value right the cut off
11:47
and then use that to put your circles or
11:49
your D and that is easy to do right once
11:52
you figure it out now now we're going to
11:54
go backward now we are given a
11:57
function how do we
12:00
write the function based on the graph
12:02
now that's harder right because now you
12:05
have
12:06
this and they want you to build
12:08
that we're going backward right first we
12:12
did we have this and we build this now
12:14
we have the graph how do we find the
12:18
function that's more delicate work right
12:21
but again if you use our cut of value
12:23
we'll be able to figure it out so let me
12:26
show you how to do that
12:33
so now I'm going to go here and I'm
12:35
going to see how many color value do I
12:37
have I need to start with those C
12:40
values how many do I have three are you
12:43
sure it's
12:44
three right see I have a gap here any
12:47
time you have a gap you have a c value
12:48
right so what's my first C value
12:51
here if I look what is
12:55
it it's always on the x axis is what one
12:58
one
12:59
right one because this is one right and
13:03
what's my second col value two two two
13:07
right because it's right here does that
13:10
make sense to everybody want to go over
13:11
it
13:12
again all right so to find your color
13:15
values you always look at the gaps and
13:17
the holes right you have the hole you
13:20
have a hole here and you have a closed
13:22
Circle you have a hole here and you have
13:25
a DOT right so therefore my cutter
13:28
values were right here a question for
13:32
the second one there there's two dots
13:34
how do you know which one it is this one
13:36
here yeah that's a DOT and a circle it's
13:39
always like you see Circle dot dot
13:44
Circle right so that's the only thing
13:46
that you look at you don't you don't
13:48
need to like there's nothing here so we
13:50
only look at the dot and the circle
13:52
that's going to be our cut of values
13:54
right so now we going to learn we going
13:57
to try to figure this out so let me just
13:59
go step by step right for the first one
14:02
X is one let's just start with that for
14:04
a second right so if I was to write this
14:07
function let we're going to call this
14:08
function f
14:10
ofx
14:12
okay so my first value is what when X is
14:16
what less than less than one how is it
14:19
less because I'm looking at this portion
14:21
here right I'm going to start with this
14:24
function when X is less than one I have
14:26
that does that make sense or no
14:30
all right if you look here
14:35
right where does this function end this
14:38
function here where does he end at
14:40
what on I start here right I'm starting
14:44
here I'm going all the way to what one
14:46
one right so these functions end at one
14:51
correct it starts anywhere here you see
14:54
that that the the arrrow is pointing
14:56
that the function keeps going right so
14:58
we start somewhere but all way one it
15:02
stops right so that mean when X is less
15:05
than one we have this function does that
15:08
make
15:09
sense if x is less than one I have all
15:13
of these and that's the function that's
15:15
being represented yes or no is it still
15:18
obscure if you don't get it then I'll
15:20
stop and I'll explain it again and again
15:21
until we get it does that make
15:23
sense who doesn't get it I'm just
15:26
confused how why do you start with the
15:29
open circle first rather than I don't
15:30
have to start with open circle I'm just
15:33
thinking I'm going from left to right
15:35
right cuz I'm I'm from
15:38
the so so for the first one would it be
15:42
like x iser one for which one for the
15:46
open circle the one that you're
15:48
doing um is X greater than or less than
15:52
less than because if you look here right
15:55
this function is where on the left or
15:56
the right the left the left so so if the
15:59
function is on the left is it greater
16:00
than one or less than one less than less
16:03
than one so the function is all the way
16:05
here so that is X is less than one right
16:09
okay so when X is less than one now we
16:13
need to figure out how to draw this
16:14
function right it's going it's not going
16:16
to be hard to find because all we have
16:19
to do is just use how many points do I
16:20
need to graph a line two two right and
16:23
remember we always use our C of value
16:25
when X is one what's y 1 2 3 4 y
16:29
five so this is one and five and I need
16:32
another one can I use this here yes
16:35
right I can use this value so when X is
16:37
zero Y is what 1 2 3 4 so 0 4 right and
16:42
now all I need to find is what
16:45
the slopes I'm going to do Y2 - y1 over
16:49
X2 - X1 right so it's going to be slope
16:53
M = 4 -
16:56
5 right over 0 0 - 1 that is -1 over1
17:04
that is 1 right so slope is 1 so Y is 1
17:09
x +
17:11
B slope is 1 x + B correct you want know
17:15
the Y
17:17
intercept I already know the Y intercept
17:19
the function is graph Crossing Y axis
17:22
here four four right so I don't need to
17:25
calculate it so that just too much work
17:26
for myself I don't want to do that so
17:29
the function is x + 4 right so we found
17:34
the first function now we need to find
17:36
the second function pay attention the
17:39
second function is right here right it's
17:41
between what one and what one and two so
17:44
I know that X is between 1 and
17:49
two right X is between 1 and two you see
17:53
that it's between one and two and I can
17:56
use these two values to find the
17:58
equation
17:59
right so I'm going to when X is 1 Y is
18:05
what x is one y is one right one and one
18:10
and when X is 2 Y is zero right two Z so
18:13
I'm going to use this two point to find
18:15
my slope and then find the Y intercept
18:18
right so I'm going to go M
18:21
= 0
18:23
-1 / 2 -
18:26
1 so m =
18:29
12 12 oh negative 1 sorry I don't know
18:33
what I was thinking thanks for cing
18:37
that maybe this is the idea maybe we'll
18:40
see it's not a big we
18:44
see1 x + B right now I need to find B I
18:48
can use this here I'm going to replace y
18:51
by 0 x by 2 and solve for b right so I'm
18:56
going to have 0 here = 1 * 2 + B so 0 =
19:02
-2 + B at 2 2 so 2 is = B so B is equal
19:08
to 2 right so my second function
19:12
is 1x + 2 now you see how we building
19:16
this right now we have one more function
19:19
to build so that last function is sp
19:23
going to be the easiest one out of all
19:24
three of them right so my cut value is
19:28
what I'm looking at this function what's
19:30
my cut of value two two so X when X is
19:33
more than what two because I'm here
19:36
right when X is more than two what's the
19:39
function it's a horizontal function do I
19:41
need to find y all this I don't need to
19:45
it's zero oh yeah so therefore the
19:48
function is yal
19:50
what y = y = 3 does that make sense y =
19:55
3 because I'm here it's a horizontal
19:57
function I don't have do I don't have to
19:59
find a y intercept I don't have to find
20:01
the slope is a horizontal function yal 3
20:06
so it's just why it's just three three
20:09
yeah three and that's it so did you see
20:12
how we built it this is again this is
20:13
delicate work so it's going to take some
20:17
you're not going to get it the first
20:18
time so easy you have to practice you
20:20
have to do a lot of practice this is not
20:22
as easy okay so what I'm going to do is
20:24
I'm going to give you some work to do
20:26
we're going to stop here for this P
20:27
function I don't to introduce the
20:29
absolute value function we just going to
20:31
do the pie wise today and that's it okay
20:34
so I'm going to have you work on a
20:36
couple of problems
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