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what are the proper steps i would need to do in order to sketch a graph
of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps
of graphing a polynomial function.
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First what you need to understand is that a sketch is not a graph.
A graph must follow to a strict definition of a function and should reproduce the function literally and precisely.
A sketch should only reproduce the major behavior and the major tendencies of a function.
For a sketch, you should reproduce
(a) the zeroes of a function;
(b) the end behavior of a function (its behavior at x ---> oo and x ---> -oo);
(c) the asymptotical behavior of a function (horizontal and vertical asymptotes);
(d) local maximums and local minimums of a function;
(e) at the zero points (x-intercepts), you should show if the sketch (the plot) intersects x-axis and changes the sign or,
in opposite, remains the sign unchanged.
If you can reproduce all these elements and peculiar properties of a function, it is just enough for the sketch.
Concretely for the given function f(x) = 0.1x^5 - 0.9x^3
you start by re-writing it in the equivalent form f(x) = 0.1x^3*(x^2 - 9) = 0.1x^3*(x-3)*(x+3).
From this form, you see
(a) end behavior: it tends to -oo at x ---> -oo;
it tends to oo at x ---> oo;
(b) it has zeroes at x= 0; x= 3; x= -3.
(c) the roots are of odd multiplicities: multiplicity 3 at x= 0;
multiplicity 1 at x= -3;
multiplicity 1 at x = 3.
Hence, at each zero, the plot (the sketch) intersects x-axis, changing the sign.
(d) the plot is in negative half-plane y < 0 at x < -3;
in positive half-plane y > 0 in the interval -3 < x < 0;
in negative half-plane y < 0 in the interval 0 < x < 3;
in positive half-plane y > 0 in the interval 3 < x < oo.
(e) There is positive local maximum in the interval -3 < x < 0;
There is negative local minimum in the interval 0 < x < 3.
I'd say, it is enough to describe the sketch for this function.
What I described in this post, is not a universal prescription and is not a universal algorithm/methodology.
You will get more experience with the time, when you will make sketches for 10 - 20 functions on your own . . .
For periodic trigonometric functions, you should be able to describe and to reproduce their midlines; their amplitudes, periods;
horizontal and vertical shifts; horizontal and vertical compressing or stretching coefficients.