Graphs the equation of a curve

Graphs the equation of a curve

We return to x and y, representing the point M. Let x and y be connected by a functional relationship This means that, on varying x (or y) arbitrarily, a corresponding value of y (or x) can be found each time. Every such pair of values x and y corresponds to a definite position of the point M on the plane XOY; when the values vary, the point M moves over the plane and thus traces out a certain curve (Fig. 4), which is called a graphical representation (or simply a graph or diagram) of the functional relationship concerned.

If the relationship is given analytically as an equation in explicit form:

or in implicit form:

we call this the equation of the curve, whilst the curve is the graph of the equation. A curve and its equation are simply different expressions of the same functional relationship, i.e. all points, the coordinates of which satisfy the equation of a curve, lie on this curve, and conversely, the coordinates of all points lying on the curve satisfy its equation. If the equation of a curve is given, the curve itself can be constructed more or less accurately on a sheet of graph paper (more strictly, any desired number of points lying on the curve can be constructed); the more points are taken, the more evident becomes the shape of the curve. This method is called plotting a curve. The choice of scale is important in plotting curves. Different scales can be chosen for x and y. The plane is taken as a sheet of paper, ruled into squares or rectangles, depending on whether the scales of x and y are the same or different. The reader is recommended at this point to plot some curves of simple functions, and to vary the scales of x and y.

 the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek