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.