Graphing Polynomial Functions

Key Questions

  • A polynomial function in standard form must look like:

    #f(x)=a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+...+a_2x^2+a_1x+a_0#

    where #n in NN# and #a_i in ZZ, i in {0, 1, 2, ..., n}#

    If you are not familiar with the notation, #NN# is the set of natural number, and #ZZ# is the set of integers.

    Examples of polynomial functions in standard form:

    #f(x)=-4x^5+7x^3-6x^2+4#
    #g(x)=3x^4-4x^3+6x-9#

    Examples of non polynomial functions:

    #h(x)=4x^7-3x^5+sqrt(2x)-5#
    #j(x)=-7x^3-2x^2+5/(x^3)#
    #k(x)=4.5x^5-3x^2#

    I'll leave it to you to figure out why they are non polynomial functions.

  • Answer:

    This is quite a broad question.
    Tips below.

    Explanation:

    Let #f(x)# be a polynomial of #n^(th# degree with real coefficients.

    To plot the graph of #f(x)# the following points are useful.

    (i) Find the real zeros of #f(x)#, if any.

    Set #f(x) =0# and solve for #x#.
    The real zeros are points on the #x-#axis.

    (ii) Find the #y-#intercept.
    Find the point #f(0)#. This is the intercept on the #y-#axis.

    (iii) Find the turning points of #f(x)#, if any.

    Set #f'(x) = 0# and solve for #x#. (Say, #barx#)

    Then,
    where #f''(x_i)<0 -> f(x_i)# is a local maximum value.
    where #f''(x_i)>0 -> f(x_i)# is a local minimum value.
    where #f''(x_i)=0 -> f(x_i)# is an inflection point.

    (iv) Plot points.

    Outside of the above simply compute #f(x_j)# and plot points #(x_j, f(x_j))# as necessary to complete the graph.

    I hope this helps.

Questions