Professor Dave here, let’s graph some circles.
In algebra, there are so many things that we could graph, but there is one class of
figures called conic sections that will be very important.
First, let’s learn what this term means.
Take this three-dimensional right circular double-napped cone.
We can create different two-dimensional figures by taking a plane and swiping through the
cone in different ways.
If the plane goes from one side of the cone to the other, and is parallel to the base,
we get a circle.
If the plane does the same thing but is not parallel to the base, we get an ellipse.
If the plane cuts from the side of the cone down through the base, we get a parabola.
And if the plane cuts all the way from one base to the other, we get a hyperbola.
So these are called conic sections because they are literally different cross sections
of a cone.
We can also describe these using a parameter called eccentricity, which is a term that
describes how much a conic section deviates from being perfectly circular.
For a circle, this value is zero.
For an ellipse, it’s between zero and one.
For a parabola, it’s precisely one, and for a hyperbola, it’s greater than one.
Let’s learn how we can graph these different figures one at a time, starting with circles.
As we know, circles are defined as the set of all points in a plane that are equidistant
from a fixed point, which we refer to as the center of the circle.
This distance from the center to any point in the circle is called the radius of the circle.
If we place the circle in the coordinate plane, we could say that any point on the circle
has coordinates of XY, because any point on the circle contains X and Y values that satisfy
the conditions of the circle.
We can also label the center as the point HK.
So how can we algebraically represent the set of points in the circle?
Well we just learned how to use the distance formula.
Again, that looks like this, and it says that when we examine any two points in the coordinate
plane, the distance between them will be the square root of the square of the difference
between the X values plus the square of the difference between the Y values.
This applies here, because any point on the circle has coordinates XY, and the center,
which is what we are measuring the distance from, will always be HK, so we get the square
root of X minus H quantity squared plus Y minus K quantity squared.
For any X and Y values that satisfy this expression, we get some radius of the circle, represented
by R. All we have to do is square both sides, and we get the standard form for the equation
of a circle, with three terms, all squared.
We can use this understanding to graph a circle given an equation.
Let’s start with something simple like X squared plus Y squared equals nine.
In this case, there are no H and K terms, so H and K must both be zero, meaning that
the center of the circle is at the origin.
Then, R squared is nine, so the radius must be three, and there’s our circle.
Now let’s look at one with H and K terms present.
Here, the only thing we have to remember is that it’s X minus H and Y minus K, so if
we have X minus two and Y minus three, the center will be at two, three, both positive,
because H equals two and K equals three, according to the equation.
If we had X plus two and Y plus three, that would be the same as X minus negative two
and Y minus negative three, so in that case, the center would be at negative two, negative three.
We can also go the other way around and derive the equation of a circle from its graph.
Look at this one here.
All we need to do is locate the center to get the H and K values, and then find the radius.
This is easy if we just go in a direct horizontal or vertical direction from the center and
see how far it is to a point on the circle.
Don’t forget to square this, since we need R squared.
So this will be the equation for this circle.
A lot of circles will have graphs and equations that are this simple, but sometimes we have
to get clever.
Say we have something like X squared plus Y squared plus four X minus six Y minus twenty-three.
If we look carefully, we will notice that this can be rearranged to be put into standard form.
We can just bring twenty-three to the other side, and then group the X and Y terms separately.
Earlier in the course, we learned how to complete the square, and that’s exactly what we will
do here, but twice, once per variable.
If you need a refresher, go back and check out that tutorial now, otherwise we simply
notice that for the X terms, we can add four to get a perfect square, making sure to add
four to the other side as well, and now this term can become X plus two quantity squared.
We can also see that for the Y terms, we have to add nine, again making sure to add to the
other side as well, and again we have a perfect square, which allows us to express this as
Y minus three quantity squared.
Add up the numbers on the right, and we’ve got a circle in standard form, with its center
at negative two, three, and with a radius of six.
So now we know how to represent circles graphically, how to write their equations in standard form,
and how to manipulate an equation in order to get it into standard form, so that we can
graph it.
Circles are relatively simple compared to the other conic sections, so let’s move
forward and check out a trickier one, the ellipse.
But first, let’s check comprehension.