What is the Difference Between a Quadratic Graph and a Semicircle Graph?

[Skhandelwal]

I am in AP Calculus but I think I just understood what a quadratic graph is and I wanted to make sure w/ you guys that I came up w/ the right explanation. I believe a quadratic is a semicircle(only y=x^2, others are semi elipse). The only diff. is that it never gets to complete its semicircle like zeno's paradox. Let's take the graph y=x^2 for example. Its origin is the bottom of the semi circle. When the graph from both sides achieves infinity, its instantaneous slope will become zero and it will be like the side of a circle. What do you think?

One other thing is bothering me, Since quadratic graph gets slower toward approaching horizontal infinity and therefore gets faster approaching vertical infinity, it doesn't make sense to my mind that it will approach horizontal infinity. It is sort of like Zeno's paradox that even though, it will keep traveling horizontally, since it will keep getting slower, there will be a limit to the amount of distance it will cover. Here is the page on what I am talking about relating to zeno's paradox. http://en.wikipedia.org/wiki/Zeno's_paradoxes
From there, I am qeoting the following: if the distances are always decreasing, the time is finite. What do you think?

 

[cliowa]

I am in AP Calculus but I think I just understood what a quadratic graph is and I wanted to make sure w/ you guys that I came up w/ the right explanation. I believe a quadratic is a semicircle(only y=x^2, others are semi elipse). The only diff. is that it never gets to complete its semicircle like zeno's paradox. Let's take the graph y=x^2 for example. Its origin is the bottom of the semi circle. When the graph from both sides achieves infinity, its instantaneous slope will become zero and it will be like the side of a circle. What do you think?

I believe you shouldn't look at the graph of a quadratic function as a semicircle, nor as an ellipse, because it is not. It might somehow look a little like it at some places if you look the right way, but hey, what doesn't look like some part of some circle if you look at it in some particular way?

There actually is a name for what you called a "quadratic graph" (notice however that the graph is not quadratic, it is a quadratic function the graph of which you're studying). It's called a parabola.

When you're given y=x^2 you can study it's behaviour and think about what happens if you let x approach infinity and so forth. This can be done without any reference to a circle.

One other thing is bothering me, Since quadratic graph gets slower toward approaching horizontal infinity and therefore gets faster approaching vertical infinity, it doesn't make sense to my mind that it will approach horizontal infinity.

I don't understand a single word you're saying. What do you mean by things like "the graph get's slower"? Formulate it this way: If you move along x with constant speed and look at the changes in y, then ... (if this is what you're trying to say).

It is sort of like Zeno's paradox that even though, it will keep traveling horizontally, since it will keep getting slower, there will be a limit to the amount of distance it will cover.

You need to be more precise: What covers what distance while what does what?

 

[CRGreathouse]

The parabola y = x^2 is defined for all real x, and so in particular for all x > k for some (any!) fixed k. I think this is what you mean when you ask about it 'reaching the horizontal infinity'. Similarly, there is an x such that y > k for any fixed k; this is what I think you mean when you say it 'reaches the vertical infinity'.

Compare this to

$$y=10-\frac{10}{|x|+1}$$

which 'reaches the horiontal infinity' but does not 'reach the vertical infinity' in the sense defined above.

Does this make sense? If so, what do you think about the function $y=\log\log x$?

 

[CRGreathouse]

When the graph from both sides achieves infinity, its instantaneous slope will become zero and it will be like the side of a circle. What do you think?

The slope approaches infinity, not zero, as x tends toward infinity. In the extended real line, $f'(\infty)=\infty$.

 

[Data]

One other thing is bothering me, Since quadratic graph gets slower toward approaching horizontal infinity and therefore gets faster approaching vertical infinity, it doesn't make sense to my mind that it will approach horizontal infinity. It is sort of like Zeno's paradox that even though, it will keep traveling horizontally, since it will keep getting slower, there will be a limit to the amount of distance it will cover.

Graphs don't get "slower" or "faster." Graphs don't even know what time is (or distance, for that matter)! A graph is just a set of points. It's a mathematical construction and has little if anything to do with the real world.

If you let $f(x) = x^2 \ \forall x \in \mathbb{R}$, then the set $\{ (x,x^2) \in \mathbb{R}^2 | \ x \in \mathbb{R} \}$ is the graph of $f(x)$. If you want $x^2$ to get large, then $x$ has to get large too (just not quite as large).

(I'm using your terminology and calling these things "graphs," but you should really call them something else [maybe "function plots?"]. A graph is actually something completely different [see graph theory].)

 

[radou]

...
(I'm using your terminology and calling these things "graphs," but you should really call them something else [maybe "function plots?"]. A graph is actually something completely different [see graph theory].)

It is enough to call it 'a graph of a function', when the context is perfectly clear, as it is here.

 

[Skhandelwal]

oops sorry, I made some typos, first of all, I mean to say that when the graph reaches infinity vertically, the slope becomes undefined, NOT 0. For the second question, I got it myself. To me, it looked like as quadratic is increasing vertically, it is slowing down horizontally, but actually it is constant horizontally, duh, those are the input values. Just relatively, looking at the graph, it looked like it. I got it now.

The only question now remains is that mathematically, I understand what a parabola is like but when I imagine it, it either looks like part of the circle or a line followed by a curve to me. I just can't imagine something curving and increasing at the the same time b/c when something is concave up, it is curving on the inside, which is how an ellipse is so I just can't imagine something curving on the inside and increasing on the outside. See what I am having problem w/?

 

[HallsofIvy]

The graph of x² is a parabola- neither a circle nor an ellipse.
A circle has eccentricity 0, an ellipse eccentricity between 0 and 1, a parabola has eccentricity 1. (A hyperbola has eccentricity > 1.)

"Curving on the inside and increasing on the outside"? "curving" upward only means that it is not only increasing but increasing faster.

 

[Skhandelwal]

Yeah and curving upward also means that it is increasing upward and decreasing horizontally. Do you have an analogy that might help? For instance, when I try drawing a parabola, and I draw it too long, my lines start getting straight.

 

[CRGreathouse]

Print a standard-looking parabola and trace it sevral times with smooth arm movements. Get a feel for how it's actually increasing.

On the flip side, consider y=sqrt(x) for x >= 0; this is the 'horizontal analogue' of the parabola. (Actually, it's just half a parabola, rotated 90 degrees.)

 

[Robokapp]

Well, quadratic graphs are studied in Algebra 2 but Circle Formula is introduced in Geometry 1. it's (x-h)^2+(y-k)^2=r^2 meaning it's in fact a square root function for a semicircle, not a quadratic. I had a difficulty understanding this also, in the area close to vertex it looks similar...however it never "rounds up". A semicircle ends in points that are non-differentiable due to tangent with infinite slope (vertical tangent) while a parabola is differentiable at all points.

I think that's the main difference for ... eye-sight measuring.