Graphing Polynomial Functions

Key Questions

  • What is a polynomial function?

    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.

    Amory W. · · Sep 13 2014
  • How do you graph a polynomial function?

    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.

    Alan N. · 1 · Aug 12 2018

Questions

Polynomial Functions of Higher Degree