Welcome to a lesson
on graphing the quadric surface,
the elliptical paraboloid.
As mentioned in the previous videos,
the degree and the sign of the degree two terms
as well as which terms are present help determine
which of the six basic quadric surfaces is given.
Then, to graph a quadric surface is it often helpful
to graph the xy-, xz- and yz-traces.
These are the intersection of the surface
with these three planes.
So to determine the xy-trace, we'll set z to zero.
To determine the xz-trace, we'll set y to zero.
To determine the yz-trace, we'll set x to zero.
And it any of these traces don't exist
or are a single point,
it's often helpful to find a trace in a parallel plane.
Let's take a look at a form of an equation
that produces an elliptical paraboloid.
Notice there are two degree two terms present
and one degree one term.
Also, if this equation was set equal to zero,
the linear term would be negative.
One trace will begin the ellipse,
and two traces will be parabolas.
And the axis will be parallel to the degree one variable.
Let's go ahead and take a look at an example.
First, we need to recognize that this does fit the form
of an elliptical paraboloid.
And now we'll go ahead and find the three traces
for this given equation.
So to find the xy-trace, we'll set z equal to zero.
That would produce the equation zero equals
x squared over four
plus y squared over two.
This looks like it would be an ellipse,
but if it's equal to zero,
the only point that would satisfy this equation
would be the origin when x is zero and y is zero.
So the xy-trace is actually just the point at the origin.
Now while that's a little bit helpful,
it would be more helpful to determine
a trace that would be parallel to the xy-trace.
So if we let z equal one,
we could change this zero to a one,
and now we could see that the trace
of z equals one would be an ellipse
where the major axis would be horizontal
and the minor axis would be vertical.
Let's go ahead and sketch this as our trace.
So we'd have a squared equals four,
so a is equal to two,
so we'll plot a point two units to the right
of the origin and two units to the left of the origin.
Notice this is a scale by twos.
Now here we have b squared equals two,
so b would be the square root of two,
which is about 1.4, so we'll plot a point somewhere in here,
above the origin and one below the origin.
So this tells us that any trace where z is greater than zero
would be an ellipse.
Now to determine the xz-trace, we'll set y equal to zero.
That'll give us z equals x squared over four.
And this'll be a parabola,
so let's go ahead and plot a few points
to determine the graph.
Zero, zero satisfies the equation,
and when x is two, we'd have two squared divided by four,
that would be one, so the point two, one is on the graph.
And so is the point negative two, one.
And now we can see we have a parabola, opens up like this.
And to determine the yz-trace, we'll set x equal to zero.
That'll give us the equation z equals y squared over two.
Again notice that we have another parabola.
Now when y is equal to two,
we'll have two squared divided by two.
So the point two, two is on this graph,
as well as the origin and the point negative two, two.
So we have another parabola, but it's not quite as wide.
So the idea here is we can put these traces together
on the xyz coordinate system
to produce the elliptic paraboloid,
but we'll go ahead and use graphing software to do this.
If we were to graph this, it would look something like this,
and you can see the yellow, red and green traces,
even though this green trace looks like it would be
at z equals five rather than z equals one,
as we found.
Let's go and take a look at this graph on Maple
so we can get a better look at these traces.
Here's our elliptic paraboloid.
If we take a look at just the xy plane,
looking straight down at it,
here you can see the elliptical traces.
We can also see that the point zero, zero
as the true xy trace should be right here.
Now, if we look at just the xz plane,
would look something like this.
And again you can see the parabola,
or the parabolic trace.
If we take a look at the yz-trace,
at this angle,
notice we have another parabola,
but now it is narrower than the xz-trace.
So here is our elliptical paraboloid.
Okay, that's going to do it for this video.
I hope you found this example helpful.
Thank you for watching.