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Algebra II: High School23 chapters  203 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Picture an ice cream waffle cone right side up. Now picture another one directly underneath it that is upside down. Now take a knife and make a cut through it. This is what we call a conic section, the shape that results from cutting though a cone.
Try cutting it in different ways. Notice how the shape that results changes depending on how you cut it. If you cut it straight across, you get a circle. If you cut it at an angle, you get an ellipse, which is a circle that has been stretched out. If you cut it parallel with the edge of the cone, you get a parabola, which looks like an arc. You get a hyperbola, or two arcs, if you cut so that you go through both the top and bottom cones.
Each of these shapes has a unique formula that identifies it. You will learn how to identify these different equations in this video. Let's get started now with circles.
We get circles when we cut our cone straight across. The formula for our circles has a standard form of
An example of a formula that will give us a circle is x^2 + y^2 = 1. Notice how we have the x^2 part and the y^2 part as well as a plus separating them. We also have the radius part, which is always positive. If you see these, then you are looking at the formula for a circle.
The formula for an ellipse, which is a stretched out circle, is almost the same as a circle. The standard form for an ellipse is
An example of an ellipse is (x  1)^2/4 + (y  4)^2/9 = 1. What we look for here is the x^2 part with a number underneath and the y^2 part with a number underneath. If one of the radiuses is 1, then you won't see a number there for that radius. We are also looking for the equals 1 part as well.
Next comes our parabola, which is an arc. The standard form for our parabola is
An example of a parabola is x^2 = 8(y  2). Our identifying point for a parabola is that if our x is squared, then our y is not, and vice versa.
Now, last but not least is our hyperbola, which looks like two arcs back to back with each other. Our hyperbola has a general form of
An example of a hyperbola is (x  4)^2/4  (y  1)^2/16 = 1. What are we looking for here? We are looking for our x^2 part with maybe a number underneath and our y^2 part with perhaps a number underneath. We also look for the y^2 part being subtracted from our x^2 part. All of this will equal 1.
I know this looks an awful lot like an ellipse, but it does have a difference. What is that? While an ellipse has a plus, our hyperbola has a minus between its parts. Remember this important difference!
Now let's review. We've covered quite a bit. All the formulas we covered are formulas for the shapes that come from cutting a cone. We call these shapes conic sections. We have circles when we cut the cone straight across. Then we have an ellipse, a stretched out circle. We also have a parabola, an arc, and a hyperbola, two arcs back to back. We can summarize our standard forms and examples in a table like this:
Conic Section  General Form  Example  

Circle


x^2 + y^2 = 1  
Ellipse


(x  1)^2/4 + (y  4)^2/9 = 1  
Parabola


x^2 = 8(y  2)  
Hyperbola


(x  4)^2/4  (y  1)^2/16 = 1 
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Algebra II: High School23 chapters  203 lessons