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Welcome to another edition of Math Tutorials...
In today’s edition, we’re going to take a look at Graphing Piecewise-
Defined Functions...
And here’s an example of a Piecewise-Defined Function...
Our instructions here are to sketch f(x)...
One thing to remember about this: If you ever get confused by
the notation…When it shows you more than piece, what it means is
there’s more than one equation. One equation applies for a certain
part of the graph; and another for a different part.
So this is telling us:
f(x), or y, is given by y = 5 - x
is one of our equations, and it applies if x 2.
So the way I always start is to take a look at the
--what I call the “switchover point”--when does it switch from
one equation to the other? The first equation applies for
x 2…so 2 is the
switchover point, OK? So...
mark x=2 with a dashed vertical line here.
This is not an asymptote, if you know what that is from rational
graphing. It’s just a mark here to indicate to myself where
it switches from one of these equations to the other.
Looking at x2,
the equation is y = x + 1. So, over here
it’s y = x + 1.
So that’s always the first thing I do: I divide up the regions
and figure out where one equation applies and where the other.
Now, if you look at the forms of these equations, each one is
a line. This is y=mx+b.
And this is it switched in order--you may be used to x first--
y = 5-x is the same as y = -x + 5, which is y=mx+b, so it’s a line as well.
Both of these are parts of lines, and one of the things
that’s characteristic of a line is that I only need to know two
points on the line to graph it. So that helps me know what I should
be doing in order to sketch these pieces.
With that in mind, let’s go ahead and start graphing
these different pieces. I’m going to
just draw one at a time. Let’s start with the LEFT one, y = 5 - x.
You can do this a number of ways, but I’m just going to
pick points since most of us are pretty comfortable with that.
Just make an xy-table and pick a few values of X.
Then plug them in and see what we get. Keep in mind, there is a
restriction: because y=5-x
ONLY over here, to the left of x=2,
then the X’s I can pick have to be
over here. Don’t just pick any X.
I don’t want X to the right. I want X’s over here (LEFT of dashed line)
So any X’s over here will do. But I want to also
in particular x=2 because I want to know
where the line ends. So I’m going to pick
X = 0
Put 0 in for X here…5 - 0 = 5, so
it goes through (0,5), right there...
Then I’m going to pick X=2 because I want to know
where it stops. Put X=2
and 5-2 = 3, so
it goes through (2,3), right there, on my dashed line.
Because I have a line here
that’s it. That’s all I need. That tells me the full
shape. So my line is going to go this way...
Like so...
OK? So that’s the first piece. That’s really all there is
to drawing the first piece of this
graph. Now we’re going to do a similar thing for
the second piece over here. I’m going to make a table...
for Y=x+1. And, even though, technically
if you look up here, it’s not defined for X=2,
--that means it’s going to be OPEN at X+2--I want to know where
that open circle is going to be, so I’m going to pick X=2.
When I put X=2,
right here, I get Y = 3.
So, if you go (2,3), that’s right where the other piece ended...
So I have an open circle right there, but it connects
It’s open on my new piece, but filled in on the first piece.
So I’ll just draw an open circle there for the moment. And then
pick some other X-value over here, on this side, for x > 2.
So maybe like X = 3 or 4…I’ll pick X = 4.
Put X =4 in here, and I get 5...
So it goes through (4,5).
Then I’m just going to draw a line that
connects those two dots.
From here to here...
And that’s our picture.
In the final picture, some ask whether to show the open circle here...
You don’t actually, because these two pieces together
form one graph. And while, it’s open for the right piece,
it’s filled in by the left piece, and these actually connect to each
other, forming one V-shape. So I don’t really need that. It’s just a V-
shape with no gaps in it. OK?
So that’s the idea of sketching a piecewise-defined graph. I start
by figuring out where it switches from one piece to the other
Here it’s at x=2. I draw a dashed line
there. Then I identify which graph (equation) applies to which side
of that dashed line. And I just graph them individually.
In this case, since they were lines, it was easy to use a T-Table to
do that. OK...
So this completes our tutorial on graphing
Piecewise-Defined functions...