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Why does a parabola close at infinity?
A science teacher made this statement and I didn't quite get it.
Thanks!
A science teacher made this statement and I didn't quite get it.
Thanks!
HallsofIvy said:What your teacher was thinking about is probably this:
Start with an ellipse with one focus at (0,0), the other on the y-axis (at (0,y), say). Imagine "stretching" the ellipse so that, while the first focus remains at (0,0), the second focus moves along the y-axis with y getting larger and larger. You will find that the eccentricity of the ellipse increases- getting closer and closer to 1.
matt grime said:you must have heard of people saying parallel lines meet at infinity? it's the same thing.
A parabola closing at infinity refers to the behavior of the curve of a parabola as it extends infinitely in both directions.
Parabolas close at infinity because the slope of the curve approaches zero as it extends infinitely in both directions. This creates a "closed" shape that does not have an end point.
The behavior of parabolas closing at infinity is important in many mathematical applications, including optimization problems, engineering, and physics. It also has useful geometric properties, such as being the shape of satellite dishes and suspension bridge cables.
To graph a parabola closing at infinity, you can plot points on a coordinate plane or use the general form of a parabola equation, y = ax^2 + bx + c, to determine the direction and shape of the curve.
Yes, there are other types of curves that close at infinity, such as hyperbolas and ellipses. These curves have different characteristics and equations, but they also exhibit the behavior of extending infinitely in both directions.