Relation, Function and Equation… Cartesian Plane and Graph

When we start to learn about a curve, we get to know about an arbitrary point (x, y) which traces the path of the curve in accordance with an algebraic equation [which is given].

For example, when we have to define a curve, say, a circle, we try to establish a mathematical expression which generates that circle [i.e., an algebraic equation which the arbitrary point (x, y) traces] We do so on a plane which acts as a reference system for the curve [or for set of all points satisfying the algebraic equation]. This reference system is called as Cartesian Co-ordinate System.

Mutually perpendicular axes of reference X-axis and Y-axis intersect at O - the Origin. Every Curve tracing is done with respect to the origin.

So, in Cartesian Co-ordinate System, we draw a plane figure [locus of a point] which satisfies the given equation. Well, that equation is a relation between x and y. For a circle of radius ''r'' with center at point (0,0) i.e., origin the equation will be x^2 + y^2 = r [simply found by using distance formula. Hence we try to find a relation between arbitrary point (x,y) and center of the circle, say, (h, k) here the origin, by means of radius ''r''. We do this as we know the property of circle that all points on its circumference are equidistant form center of the circle and that distance is radius.] Hence we generate the curve.

When it comes to graphing a function, we say there are two sets R and R [R - set of Real numbers], we take the cross products which results in a plane like Cartesian Plane. But is it the same or different? It has X and Y co-ordinates. We can draw a curve here too. We just don't relate the points, we relate them such that y is the unique image of x. A function has its domain and range [set of numbers which obey what the function says, i.e., the points where function is defined.. its real]

What is the difference between the reference planes where we trace a plane figure and plot graph?

When we are asked to plot the graph of circle we discussed above, it turns out to a semicircle lying on one side of X axis. [as we are plotting a function, it should pass the vertical line test!]

But what makes them different? The reference Planes.. are they different? I know one thing that Graphs are relative and we plot how y changes / depends on x. But in case of tracing a curve we simply do it by a known algebraic equation that describes it.

While doing analysis of a physical situation what do we use? I mean, say. a particle is moving and we have got to represent its motion. A position - time graph says where a particle is / was at a given time. It doesn't say actual path traced by the particle. But to get that, we should trace a curve using analytic geometry i.e., a position v/s position curve.

Am I right.

I have already messed up what I really intended to ask. It is as if I know the answer / concept but can't convince myself / put them together or find link..

If you are getting me, please answer. Its kinda silly and is worth being reported as a wrong or bad question. I accept that.

Thank you anyways.

• Thank you. Clearly a relation is a set of all points [i.e., all (x,y)] which satisfy a rule, here, x^2 + y^2 = r and function is a special type of relation such that for every x there is a unique y. I seem to understand this at basic level but I have had misconception about Cartesian plane and graph. Both of your answers cleared my doubts on it. Thanks. – Ahana Oct 1 '15 at 5:14
• But, let us consider the case of analysing the motion of a particle. If we are told to plot its position at various times, we plot graph which describes a simple relation [actually function] between x and y such that y depends on x. So it turns out to be position time graph. So, I thought whenever we are told to plot a curve involving all x and y such that y depends or varies with x, then it is a ''graph'' [function]. – Ahana Oct 1 '15 at 5:22
• So, that graph doesn't show actual path traversed by the particle but variation of position with time. You can also consider the example of an oscillating simple pendulum. It moves to and fro but if we plot position time graph, it will be a sinusoidal function of time. Graph doesn't show actual path but variation of things taken into account [ofcourse those things, i.e., x and y will have a relation y = f(x)] – Ahana Oct 1 '15 at 5:27
• But if we are to plot its motion in such a way that we trace the curve it actually traverses, then may be we find an algebraic expression [locus of a point] by observing what path the particle traces and plot it in Cartesian Plane. For example position v/s position curve for a particle engaged in circular motion gives the curve ''circle''. Now how am I to put them together and understand the intricacies of Cartesian Plane and Graph? I couldn't help. Thank you. – Ahana Oct 1 '15 at 5:32

Hmm...

I think she is asking clarification between a reference system which is simply used to describe a particle in 2 or 3 dimension [its position, path or displacement] and graph where we plot variation of a quantity with respect to the other.

Well, we made a reference system so that we can strictly define a particular point or event in space. But only the co-ordinates of a point are not sufficient, so the vectors are the tools which define a point, a line or a plane in space or a 2 dimensional system.

You should first study the Cartesian Co-ordinate system to know how points are plotted in a plane. A point is generated by a simple relation between its co-ordinates x and y. It may be random or described by an algebraic equation. So, if there happens to be unique y for x, it will be a function [for all x belonging to the domain of definition]

This is how we know about plotting a curve [which is basically showing which x is related to which y by a rule] It can be a relation or a function. This is where a basic math about this ends.

Then lets talk about Vectors and 3 Dimensional Geometry. There you are, to again learn how to mark a point, know its co-ordinates [x,y,z], plot a curve etc use of vectors makes it all easier.

To put it all together, "There is nothing called ABSOLUTE " when we are measuring anything. So every measurement is done relative to the other. By looking at your question I feel like you are a physics student who has trouble in applying math to physical world. So, let us make things clear one by one.

Like I said, measurement is relative. You can't just say a particle is at ''Rest".. it is at rest with respect to some other quantity. If anything is moving that means it is moving [changing position with respect to time or your position] relative to some quantity. So, there was a need to create a reference system. Thus arose Cartesian Co-ordinate system. It has mutually perpendicular axes X Y and Z. with an origin O with respect to which we can make measurements.

In math we JUST learn how to plot an algebraic equation i.e., a relation [sometimes] and a function. So, using terms like variation / dependence of a quantity makes MORE sense in physics.

We know some math and we effectively try to apply it to the physical quantities or changes we see by establishing a mathematical relation between objects. So, whether your curve is talking about actual path traversed by a particle or variation of its position w.r.t. time is a simple matter of what you are plotting in a Cartesian Plane. like position - time or position - position variation. As the X O X', Y O Y', Z O Z' are just axes of a REFERENCE system RELATIVE to which you can make measurement. So, what you plot tells you whether its a relation or function [in math] or actual path [position - position curve... Well while studying things like position vector or displacement vector there will be elaborate explanation for this. Also study 3 D geometry. How a particle moves or traces its path in space is better understood by this. ] or variation of its position w.r.t time. So is true for the case of simple pendulum. It all depends on what you are measuring relative to the reference system you have. Don't think a graph and Cartesian plane are different. We just want a reference system to make a sensible measurement which will be true regardless of time etc.

"No measurement is ABSOLUTE"

Hope this works for you.

• Thanks.. Cleared all doubts. I really had messed it up.. "No measurement is ABSOLUTE"...! I got it ! Thank you so much.. ;-) – Ahana Oct 2 '15 at 6:14

Okay...

A "graph" is a collection (x, y) points that are plotted. (We aren't tracing a (x,y) point; we are plotting a collection of several different (x, y) points.) If all the points plotted have a relationship between their x-coordinates and their y-coordinates, all the better. The relationship can be anything. It could random or it could be nothing we'd reasonably call a "relationship".

When we are introduced to the idea of a function, we are taught to think of it as a type of rule that takes a number as an input an processes it into another number. But a subtle way to think of it is as a set of pairs of numbers; the set {[x, f(x)]} (where each x has at most one f(x)). So the function f(x) = x + 2 can be thought of as {[0,2], [1,3], [1/2, 2 1/2], [$\pi$, $\pi$ + 2]......} = {[x, x+2]| x in R}.

Well, pairs of numbers can naturally be thought of as 2-dim points. so if f(x) = whatever is equivalent to {[x, f(x)]} it makes sense to plot all these as points on a graph. Thus the graph is now a collection of (x, f(x)) points.

Now normally for most graphs the relationship between the x-coordinates and the y-coordinates of a point could be anything. For a function, there can only exist one y = f(x) value for each x value. Hence the "vertical line" rule.

Um. Does this help? It was a pretty subtle question.

When you plot the curved defined by the equation $$x^2 + y^2 = r^2, \tag1$$ in effect you find a set containing all the pairs of numbers $(x,y)$ that satisfy equation $(1)$, and the curve is the set of all points whose Cartesian coordinates $(x,y)$ (in the chosen system) belong to that set.

A set of pairs of numbers like this defines a relationship. In this particular case, we say $x$ is related to $y$ if $x$ and $y$ satisfy equation $(1)$, that is, the pair $(x,y)$ belongs to the set that defines the relationship.

A function is just a special kind of relationship in which no single value of $y$ appears in any of the pairs $(x,y)$ more than once. (This is another way of describing the vertical line test.)

So when you write $y$ as a function of $x$, for example, like this, $$y = f(x) = \sqrt{r^2 - x^2},$$ you are simply defining another relationship between $x$ and $y$. You can use that equation to define a curve in exactly the same way that you used equation $(1)$. The only difference is that since in this case $y$ is a function of $x$, your curve will pass the vertical line test, which is clearly not true for every equation that defines a relationship between variables $x$ and $y$.