This next example requires graphing a piecewise
function. So, now, we can do this problem
both by hand and with a calculator using technology.
If I wanted to do this problem by hand, I
would have to make a table. And on this table,
I would have to put x's on one side and y's
on the other side. Now, in this particular
case, the negative 1 is going to be called
like the border basically. So, what I'm going
to do is in the middle of the table, I'm going
to put negative 1. And then before that point,
I'm going to pick numbers that are smaller
than negative 1, maybe two or three of them.
So, like for example, negative 2 and negative
3, and after negative 1, I'm going to pick
a few numbers again, so maybe 0 or 1 just
randomly, and make sure that the negative
1 makes it to the table because it's an important
part, it's actually going to be a place where
a split happens. So, to remind me that something
strange is going on at negative 1, I'm going
to right away, put a little dashed line right
here, where x would be equal to negative 1.
Because this dashed line is going to be there
to remind me that, on the one hand, I'm going
to have a graph; on the other side, I'm going
to have a different graph. So, that's where
the cutoff happens. So, that way I don't,
you know, extend the lines too far out. Now,
when you are investigating what negative 3
does, you'll have to remember that negative
3 satisfies the first condition. And so, therefore,
negative 3 has to be plugged into the first
equation. So, when I do that, negative 3 gives
me 2 times negative 3 minus 1, which has at
the end negative 7. So, negative 3 as input
gives us negative 7 as output, so down here.
And again, when I plug in negative 2, it will
satisfy the first equation again, so negative
2 goes into the first equation, and we end
up with negative 5 as a solution. So, when
x is negative 2, the y is going to be a negative
5. So, that's the second point and now negative
1 itself. Now, when I plug in negative 1 it
makes the second equation -- satisfies the
second inequality, I should say and makes
that one true. So, therefore, for the negative
1, I have to use the second equation, negative
3x plus 4, keep in mind that x is negative
1, so this will be 7 at the end. So, when
x is negative 1, the output is actually 7.
So, on the dashed line, the point is going
to be 7. So, that's going to be a solid point.
Now what about the one down here? Now if I
extend the line all the way to the border,
it's going to hit the point right there where
the point would be negative 3. So, at that
location, I'm going to put a circle but it's
an open circle. So, this reminds me that this
is where the line would have ended if it touched
the border negative 1. However, in this case,
the border negative 1 would have the point
up here at 7. Then if I do 0, 0 would make
again the second equations satisfied. So,
0 would be plugged into the second equation
to give us at the end 4. So, when x is 0,
the y is 4, like at that point. And then finally,
when x is 1, the second equation is going
to be used again. And the output is 2. So,
when x is 2, the y is going to be a 1. So,
those are the coordinates, and I plug them
in, and I connect them. And that's pretty
much it. That's how the graph is going to
end. So, the graph is made up of two lines,
one that points up and one that points down,
and there's a gap between the two lines, and
it is what it is. So, that's how you would
do it by hand. But again, you are not required
to do this by hand, you can use calculators
to do this. So, if you have a TI-83 or 84,
now's a good time to turn it on. And watch
what we're going to do next to graph this
problem. So, this is my TI-83 or 84 view.
I'm using a TI-84 plus CE in this screen,
but it works the same way for TI-83s and 84s,
regardless of which model you have. Now since
this is our first time working on a calculator,
let's go ahead and set it all up so that we
don't have any errors for the rest of semester.
So, what I'm going to do next is clear your
calculator memory. So, this way if you have
any errors or if you have any programs saved
or whatever, they're going to disappear so
that we're all starting with a brand new calculator
as if it was purchased just today and opened
out of the box. So, if you're using a use
calculator, or if you're borrowing calculator
from a friend or whatever, this will be a
good thing to do to reset the calculator memory.
And the way we do that is we go and first
turn it on with the On button, and once it's
on, we click Second and plus sign. That gets
us into the Memory menu, and if you see the
M is in blue, and if you have a TI-83, it
will be in yellow. So, anything in blue or
yellow requires first clicking the Second
button. Now on this list, there's a bunch
of commands that we can use. I'm going to
pick option number seven. On the screen, it
says Reset. Now, how can I get down here?
I can click my arrow down a few times until
I get to seven. Then once I get to seven,
I click Enter, or, you know, I could do a
seven on my keyboard, and it will take me
straight to that setting without having to
click Enter or anything. So, I'm going to
click down a few times, then click Enter.
And it will say, do you want me to reset everything,
all RAM, or the defaults? I'm going to say
all RAM. So, it's already on one. So, I'm
just going to click Enter one more time. And
it's going to ask me, are you sure you want
to do this? Because once I do this, it's going
to erase all the settings from the calculator.
So, if you have, you know, a calculator you're
using for physics and you've saved some software
in there or some apps in there, they will
get deleted once we do this. So, I'm going
to say, yes, reset everything and this is
number two. So, I can go down, and then click
enter, or I could just select two on my keyboard,
and it will do it for me. So, two, Enter,
and there it is. Your screen may look a little
bit different with some different writing,
but it's going to say, RAM cleared, on the
bottom. So, that's great. Now we've cleared
the memory. So, now let's go ahead and try
to plug the function into the calculator.
To do that, to graph anything, we're going
to go to the y menu, which is the top left
corner. Mine's have colors, yours may not
have colors. It doesn't really matter. Just
plug in the function into this y1. Now because
I have a piecewise function, I'm going to
be a little bit more careful. So, what I'm
going to have to do is type the equation to
x minus 1 first by putting parentheses around
it, so I'm going to put two and then x. A
convenient way to find x to just click this
button right next to alpha. That puts an x
on your screen, and then big old minus sign
and then 1. Now there are two minuses on your
keyboard: there's one next to the six, and
there's one underneath the three. The one
underneath the three has little parentheses
around it. That would use -- that will be
used if you want to indicate that a number
is negative. So, like negative 4 or negative
5, not when you're subtracting two values.
When you're subtracting two values, you use
the big old minus sign right next to six,
then close the parentheses. Now, because this
is a piecewise function, I have to add some
extra information. I have to pick out -- tell
the calculator that x is supposed to be less
than negative 1. So, I'm going to put new
parentheses, type x again, and then I have
to put less than. To find less than click
Second, then click on the Math button. So,
that gets you into the Test menu. It's either
in blue or yellow again, that's why I clicked
Second first So, Second, and then Test gets
you in here, and find which one is the less
than. Mine is number five. So, again, you
can either click down a few times to get there,
or you can just click on your keyboard five,
and it will put less than on your screen.
And now we need to put negative 1 because
that's the number in the condition, negative
1 is negative, and then 1. So, notice how
I used the little minus sign here. Whereas
I use the big old minus sign earlier because
earlier I was subtracting two numbers. Right
now, I'm just saying that the 1 is supposed
to be negative, then close the parentheses,
and that's the end of the first line of my
piecewise function. Now there's a second line
on the piecewise function, so I'm going to
put a plus sign, then put parentheses again,
and then type the equation on the second line,
which is negative 3, then x plus the 4, close
the parentheses, then put the condition. The
condition is x squared then equal to negative
1. So, again, I put it inside its own parentheses.
Parentheses, x greater than or equal to, so
that's Second, and then click on the Math
button to enter the Test menu and figure out
where the greater than or equal to belongs.
It's in number four. So, I can go down a few
times, then click Enter to select that guy.
And my number is negative 1. So, tiny little
minus sign, 1, close parentheses. I like to
click enter. That way, I get to the next line.
If I have to put another equation, I can put
that here, but for now, it is saved into the
calculator. So, to see the graph, all you
have to do is click on Graph. And you pretty
much see what I'm seeing on my screen. Yours
may look black instead of blue, or red or
whatever, it doesn't really matter, but that's
what the graph looks like. And from here,
we can extract some information to be able
to graph the function. We can do it in a few
different ways. So, for example, we can click
Second, and then Graph that puts us in a Table
menu. So, we can see when x is 0, the y is
4. When x is 1, y is 1. And if I go up that
table, I can see additional numbers that I
can now grab and put on my table on the paper
in order to help me graph the function. So,
I can either extract the information from
this nice, organized table, or if you wanted
to, you can go back to Graph, then click on
Trace. So, when you click on Trace, it puts
the little cursor on the screen, and it tells
you this point is 0, with a y coordinate of
4. And the nice thing about this is you can
select whatever numbers you prefer. So, like
for example, you may like the point negative
5, so you can click that on your keyboard
and then hit enter. And it will tell you when
x is negative 5, the y is negative 11. Somebody
else may say, okay, I want to know what happens
when x is negative 2, for example. And again,
it will tell you when x is negative 2, the
y is negative 5. If you click on negative
1, you'll notice that the calculator goes
to the top. That's where the closed circle
was on the paper. So, you'll know that when
x is negative 1, y is 7. So, if you want to
know what this one is, you will have to kind
of read it off on the side. It's about negative
3. That's why our open circle was around negative
3. Or you can try to do like some guesswork.
So, you can try to do maybe negative 1.0 something,
something, something. So, 1 point, for example,
001. That will put you super close but not
exactly at negative 1. So, you can see it's
approximately negative 3 on the y. So, that's
where the open circle is going to have to
be written down. So, again, that's a quick
way to do it with a calculator, and it gives
you the same exact graph.
