TopGraphing is a better way to understand mathematical expressions, functions, relations etc as they are pictorial representation of any quantity. However, it is possible to graph Polynomials with one or multi variables by hand. A polynomial can be thought of an expression representing a relationship between one or more variables and constants. Graphing polynomials can be imitated by following just few steps.
Note the sign and power of first term of polynomial to determine output of that term for extreme values of variable say “x”. These extremes are two ends of graph for a polynomial where there is no need for calculations. We can have four cases depending on sign of terms having 'x', and also whether power is odd or even. Graph approaches negative infinity when power is odd and sign is positive because 'x' also approaches negative infinity, and positive infinity when 'x' goes to positive infinity.
Next Set f(x) = 0 and solve for 'x'. Here you need to be strong with concepts of Algebra and its operations. For some polynomials, you may find f (x) = 0 for various values of 'x'.
Now you need to differentiate the polynomial. Set first differential equation equals to zero (f'(x) = 0) and solve for 'x'. This step lets you find critical points where polynomial changes direction. Then, substitute these points in original polynomial, and solve for f(x). Each of these pairs (x, f(x)) is a critical Point.
Similarly, taking second differential of x (f''(x)) and substituting each value of 'x' for which we got a critical point.
If f''(x) > 0, point is a local minima;
Otherwise if f''(x) < 0, the point is a local maxima.
However, if f''(x) = 0, it is not clear if the point is a maxima or minima.
Now when you have got points to be graphed, plot them to get pictorial view of polynomial. Then, connect points and draw ends according to first term.