Did you know that all parabolas are the same shape?
I mean this in the sense that all circles and all squares are the same shape. You can always go from one circle or square to another by rescaling. While it is obvious that all circles and squares are the same shape, it may seem counterintuitive for parabolas. I know it seemed that way to me when I first found out.
Part of the problem is that while circles and squares are of finite size, parabolas are infinite. How can you say that two things that are infinitely large have the same shape.
Here is a way of looking at it. Consider the curve y=x^2 where the scale is in mm. So we have points (20mm, 400mm) and (30mm, 900mm). If we rescale to cm, the same points would be (2cm, 40cm) and (3cm, 90cm). The points are in the exact same place, but in cm they satisfy y=10x^2. In general, rescaling by a factor of 1/k converts the equation y=x^2 to y=kx^2.
Imagine a wide parabola drawn on the ground. If you go up in an airplane, the changing view is equivalent to rescaling. The same point that was at 10 feet may now appear to be at only 1 foot. The scalings will have factors of 1/k. If the original equation of the parabola was y=x^2, what you see will be the narrower parabola y=kx^2.
If you were originally a doubter, have I managed to convince you?
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12 Answers
You’ll need to add qualifiers to your universal assertion.
The parabola for y = x^2 is not the same as the parabola for y = x^4.
@CWOTUS The equation y = x^4 does not actually form a parabola.
Look at Wikipedia‘s diagram of a parabola and consider the definition is ”...a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix). The locus of points in that plane that are equidistant from both the line and point is a parabola.
Since the distance from points in the plane from the line and the point, the only thing different about parabolas is what that distance is.
I, um, don’t even know what a parabola is, so I’m going to take your word for it that they are the same shape. Off to check out @ETpro‘s link!
How can you have gotten through middle school without solving and graphing quadratic equations? In simplest terms, a parabola is the path taken by a ball when you throw it, which is especially apparent if you throw it high and far.
@augustlan You would have learned it in geometry class in high school. If you aren’t much of a math person it could easily leave your mind after the tests. LOL.
It’s a conical section of a cone taken parallel to one of the surfaces of the cone. Wikipedia has a good example of it here.
Wikipedia says:
A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar, meaning that while they can be different sizes, they are all the same shape.
Interesting article here:
Like squares and circles, unlike rectangles and ellipses, all parabolas are similar. They cannot be “pointier” or “wider”. They all have exactly the same shape, which appears “pointier” from afar, and “wider” when looked at in the neighborhood of the vertex.
Unfortunately, many of us have misled many students by implying otherwise: we often claim that changing the value of a in the formula y=ax2 changes the shape of the parabola. In fact, many teachers believe this to be true.
Then he gives an algebraic proof.
if y = ax^2 (where a is the “scaling factor”)
then ay = a^2x^2
ay = (ax)^2
...which is the same equation (same graph) as y=x^2, with x and y scaled by the same zoom factor a. “All parabolas are similar.”
If I ever knew it, then I long ago forgot it again. Thanks, @LostInParadise, for today’s lesson in conic sections!
Hmm, I’m not convinced they’re all the same shape. Aren’t some more pointy and some more round?
I never took anything beyond Algebra, guys.
@augustlan That explains it.
To be honest, in reference to the Q, I didn’t remember that parabolas are all the same like circles and squares if that is indeed true and I studied through calculus, although geometry was a struggle for me when I took it in high school. Basically a parabola is like an arc. You plot it on a graph. The graph would have an X axis (that is the horizonatal line) and a y axis (the vertical line). The equation would tell you where to plot the coordinates, and then when you connect the dots it makes an arc.
Here is a way of simulating the airplane demonstration. View the parabola and keep zooming in on your browser. The parabola will keep getting wider.
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