The study of calculus and its applications is best understood when it is studied through the geometrical representation of the functions involved. In order to investigate the nature of a function (graph) it is not possible to locate each and every point of the graph. But you can sketch the graph of the function and know its nature by certain specific properties and some special points.
Domain, Extent, Intercepts and Origin
- Domain of a function y = f(x) is determined by the values of x for which the function is defined.
- Horizontal (vertical) extent of the curve is determined by the intervals of x (y) for which the curve exists.
- x = 0 yields the y-intercept and y= 0 yields the x-intercept.
- If (0,0) satisfies the given equation then the curve will pass through the origin.
The curve is symmetrical about
- the x-axis if its equation is unaltered when y is replaced by −y.
- the y-axis if its equation is unaltered when xis replaced by −x.
- the origin if it is unaltered when x is replaced by −x and y is replaced by −y simultaneously.
- the line y = x if its equation is unchanged when x and y are replaced by y and x.
- the line y = −x if its equation is unchanged when x and y are replaced by −y and −x.
Asymptotes (parallel to the co-ordinate axes only)
If y → c, c finite [x → k, k finite] whenever x → ±∞ [y → ±∞] then the line y = c [x = k] is an asymptote parallel to x-axis [y-axis].
Determine the intervals of x for which the curve is decreasing or increasing using the first derivative test.
Special points (Nature of bending)
Determine the intervals of concavity and inflection points using the first and second derivative test.