Distinguishing Conic Sections From Graphing Format
Let's review the patterns for each conic section:
Circle: (x-h)2+(y-k)2=r2
Parabola:![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_1.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_2.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_3.png)
Parabola:
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_1.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_2.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_3.png)
Let's note a few key differences:
To complete the square for ellipses and hyperbolas:
- Parabolas only have one squared variable. (For all the others, both the x and y are squared.)
- Circles do not have fractions/denominators. Also, they may or may not equal 1.
- Ellipses have different denominators and equal 1. There is a plus sign in between the fractions.
- Hyperbolas also have different denominators and equal 1, but there is a negative sign in between the fractions.
Let's identify a few conic sections:
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_4.png)
This is an ellipse because both variables are squared, it contains fractions, it equals 1, and there is a plus sign in between the fractions.
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_5.png)
This is a circle. You can tell because both variables are squared but there are not fractions.
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_6.png)
This is a hyperbola because both variables are squared, it contains fractions, it equals 1, and there is a negative sign in between the fractions.
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_7.png)
We know this is a parabola because only the y is squared.
Identify each equation as a parabola, circle, ellipse, or hyperbola.![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_8.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_9.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_10.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_11.png)
![](/math/calculus/images/distinguishing_conic_sections_from_graphing_format_img_12.png)
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