SOLUTION: what are the proper steps i would need to do in order to sketch a comprehensive graph of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps of graphing

Algebra ->  Graphs -> SOLUTION: what are the proper steps i would need to do in order to sketch a comprehensive graph of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps of graphing       Log On


   

Question 1192190: what are the proper steps i would need to do in order to sketch a comprehensive graph of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps of graphing a polynomial function.
Answer by ikleyn(52835) About Me  (Show Source):
You can put this solution on YOUR website!
.
what are the proper steps i would need to do in order to sketch a highlight%28cross%28comprehensive%29%29 graph
of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps
of graphing a polynomial function.
~~~~~~~~~~~~~~~~~~


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.