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This video is going to be about setting the viewing window for a graph,
and
sometimes you might be given a problem like this:
y = x-squared - 4x -165,
y = x-squared - 4x -165,
and you've got a graphing calculator and and you're told
to find an appropriate viewing window for this graph.
Now of course what you could do is you could put this into your calculator
and then you could...
Actually, if you put this into your calculator as it is
and you try to look at it in a standard window, you won't see anything.
In fact, if you want to pause this video and try it, go ahead.
Now of course what you could do is you could zoom out
and you could adjust different parts of the viewing window until you get it to
where you want it, but
sometimes it's simpler just to do it mathematically. So here's how we're going
to do it.
I've rewritten the equation and set it equal to zero. So I've got
x-squared minus 4x -165 = 0.
And then I just factored
that
quadratic
side, the x-squared - 4x -165 and I ended up
finding out that x equals -11
and x equals 15. So, you know this process.
Well remember, when I say x equals -11 and x equals 15, those
are going to be the x-intercepts
on a graph.
So I know what the x-intercepts are, and they're going to be important
when I want to actually
finding the appropriate viewing window.
I'm also going to need the y-intercept and that's very easy to find.
Let's take this original equation
and set it back
so it's equal to y.
The y-intercept happens when x is zero. So if x is zero,
then
y equals zero squared
minus 4 times zero,
both of those are just zero,
minus 165.
So the y-intercept
is going to be at
(0, -165).
(, -165).
Now I also want to know what the vertex is,
since that's a significant part of the graph also.
So I can use to vertex formula to find that. Remember the vertex
is...
the formula for that is
-b over 2a.
If I go to the original equation...
b was -4, so -b would be 4,
and
'a' is just 1, so 2a is 2.
So the vertex,
the x-value for the vertex,
is going to be at 2.
And I have to find what the y-value is going to be.
So what I'll do
is I'll take
my equation, y equals x-squared
minus
4x minus 165,
and I'll take
the x's and replace them with 2's.
So y equals
2 squared, which is 4,
minus 4 times 2,
which is -8
minus 165.
And tha's go to end up as, let's see... 4 minus 8 is -4, and -4
minus 165 is -169.
So, -169.
So, -169.
Okay, so now I know
that my graph
has these two x-intercepts
at -11 and 15.
I has a y-intercept at -165,
and it has a vertex at (2, -169).
And I want to set my viewing window.
So if I press the 'window' button on the calculator,
I'm going to have
x-min and x-max and x-scale, and then y-min and y-max and y-scale, and
by default it's usually set to x-min at -10,
x-max at 10,
x-scale at 1,
and then a -10,
a positive 10
for the y-min and the y-max and 1 for the y-scale.
Now remember,
these were the point I wanted to make sure I include in my graph. I want to include the
x-intercepts,
the y-intercept and the vertex.
So here's
what all those min and max values are about.
Let's say I've got
a viewing window for my graph.
Let's say it's something like that.
And the graph I'm going to put in
actually looks something like this. It's an
upward
opening parabola,
and
i want to make sure I include
the points
where
I have my x-intercepts.
Now I've got two x-intercepts. One is at -11, one is at 15.
So the lower one, the -11, has to be part of my graph,
which means I want to have an x-value that's at least -11 on my graph.
The calculator sets it to -10,
so I'm going to change that
to a little low than -11 in fact.
Let's make it -13.
For the x-maximum,
my other x-intercept is at 15,
So I want to make sure that my graph goes at least out to 15, probably a
little bit more.
So I'll set my x-maximum...
let's make that 17.
I'll get back to the scale in a second.
For the y-minimum,
I have to include...
let's see what kind of
y-values I had.
Well there's a y-intercept at -165. The vertex was even
lower,
that was -169.
So that means
I want to make sure my graph goes down at least as far as -169,
so I include this vertex.
Let's make a little lower.
Let's make the y-minimum be
-175.
And then
we want to go up a little bit higher than the
x-axis
for the y-maximum. I'm just gonna leave it at 10,
because if I'm going down as far as -175, I want to have a
little bit of room above the x-axis. So I'll leave that at 10.
Now what about this thing for scale?
Well, what scale does is basically it says
how often you're gonna have tick marks. You know the tick marks...
if I put some tick marks in here and I say that each tick mark is 1,
then my x-axis is gonna read
1, 2, 3, 4 and 5.
So, for the x-scale...
I'm going from -13 up to 17. That's 30 units altogether from
-13 over here to 17 over here.
If I think I can put 30 tick marks then I could leave the x-scale at 1.
one
If I'd rather have my tick marks spaced a little further apart, I could put them at
maybe 2.
You might want to go as far as 5.
I'll make it 2. Some of this is just a matter of opinion,
but you want to make sure that you can actually count the tick marks.
On the y-scale, it's going to be more important.
My y-scale goes from -175 up to
10.
That's the difference of a 185, which means if I leave the
y-scale at 1, I'm gonna have 185 tickmarks on that y-scale.
When you do that what you end up with is just a thick line
because all of those little pixels just blend together.
So if I've got that much distance,
I think I might want to
make my y-scale
10.
In other words, I'd be counting by 10's...
10, 20, 30, 40 and so on. I could even make it 20 or 25.
Again,
what your scale is set at is somewhat a matter of
opinion or aesthetics,
but make sure that you can at least use the scale
and count along it, rather than have it be one think line. Okay?
So try putting that equation into your calculator,
try setting your window
this way,
and see how it looks.
It should be something fairly decent. You might want to fool around a little bit,
like I said, with the tick marks.
But they should work.
So the basic concept is this:
when you start out with your equation,
what you want to do is find
your x-intercepts,
your -intercepts, and your vertex, because all of those things have to be included
in your final graph.
Once you've got that,
go to
the window menu on your calculator,
and make sure to set your window
so you include, on the x-axis,
the lowest value for the intercept
and the highest value for the x-intercept,
and on your y-axis, you want to include
the vertex.
If it's an upward opening parabola
the vertex will be the lowest point on the graph, so that will be the
y-min,
and
you want to include
someplace where the y-maximum would be,
and you want to make sure, of course, that you show both of the axes.
And then
adust the scale so your tick marks are readable.
Okay,
so that you do it.
Take care, I'll see you next time.