Graphs of Polynomial Functions

This page contains an applet to help you explore polynomials of degrees up to 5:
\[ f(x) = a x^5 + b x^4 + c x^3 + d x^2 + e x + f \]

by changing the values of the coefficients \( a, b, c, d, e\) and \( f \). It is not easy to draw any conclusion when you change all 5 coefficients at the same time. You can always reduce the degree (highest power) by setting some parameters to zero. For example if you set coefficients \( a \) to zero and \( b \) to a non zero value, you obtain a polynomial of degree 4. Once you finish this interactive tutorial, you may want to consider a Graphs of polynomial functions - Questions. If needed, Free graph paper is available.

Interactive Tutorials Using an App

A polynomial f(x) with real coefficients and of degree n has n zeros (not necessarily all different). Some or all are real zeros and appear as x-intercepts when f(x) is graphed.

A - Explore Real Solutions of Polynomial Equations of the Form

\[ x^n + f = 0 \]

where \( n \) is even or odd and \( f \) is a constant.

\(a \) = \(b \) = \(c \) = \(d \) = \(e \) = \(f \) =


Click the button "Plot Polynomial" to start.

  1. Set all coefficients to zero except \( a \) and \( f \).
    Write down the polynomial and its degree, examine the graph obtained. How many x-intercepts ( or real solutions to the above equation ) the graph has? Repeat for different values of a and f

  2. Set all coefficients to zero except \( b \) and \( f \).
    Write down the polynomial and its degree, examine the graph obtained. Change b and f and see how many x-intercepts the graph has? Which values of f give intercepts and which values do not give any intercepts?

  3. Set all coefficients to zero except c and f.
    Write down the polynomial and its degree, examine the graph you obtain. Change c and f and see how many x-intercepts the graph has?

  4. Set all coefficients to zero except d an f.
    Write down the polynomial and its degree, examine the graph you obtain. Change d and c and see how many x-intercepts the graph has and for what values of f

  5. Set all coefficients to zero except e and f,
    write down the polynomial and its degree, examine the graph you obtain. Change e and f and see how many x-intercepts the graph has?

    From 1,2,3,4, and 5 above, what conclusion can you make as to the number of solutions of polynomials equations of the form \[ x^n + f = 0 \]
    depending on whether \( n \) is even or odd and \( f \) is negative, positive or zero?.



B - Explore Even and Odd Polynomials


  1. Set a, c and e to zero, write down the
    polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

  2. Set b,d and f to zero, write down the
    polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

C - Zeros of Polynomials


NOTE: For the next tutorial, a repeated zero of multiplicity m is counted m times.

  1. Set \( a \) to a non zero value (polynomial of degree 5). Change all the other coefficients (non zero values if possible) so that the graph of \( f(x) \) has:
    1 x-intercept.
    3 x-intercepts.
    5 x-intercepts.
    Why do you think we can obtain only an odd number of real zeros of \( f(x) \)?

  2. Set \( a \) to zero and \( b \) to a non zero value (polynomial of degree 4). Change all the other coefficients (non zero values if possible) so that the graph of \( f(x) \) has:
    no x-intercept.
    2 x-intercepts.
    4 x-intercepts.
    Why do you think we can obtain only an even number of real zeros for \( f(x) \)?

    (The last two exercises are not easy, however they are very useful to fully understand polynomials).

D - Leading Coefficient Test


  1. Set the leading coefficient \( a\) to a positive value (polynomial of degree 5) and set b, c, d, e and f to some values.
    As x increases without bounds, does the right side of the graph rise or fall?
    As x decreases without bounds, does the right side of the graph rise or fall?
    Change \( b, c, d, e \) and \( f \) and see if the above behavior changes.

  2. Set \( a \) to zero and \( b \) (leading coefficient) to a positive value to obatin a polynomial of degree 4 and carry out the same exploration as in 1 above and 2 above.

  3. Set a and b to zero and c (leading coefficient) to a positive value (polynomial of degree 3) and do the same exploration as in 1 above and 2 above.

  4. Set a, b and c to zero and d (leading coefficient) to a positive value (polynomial of degree 2) and do the same exploration as in 1 above and 2 above.

  5. Set a, b, c and d to zero and e (leading coefficient) to a positive value (polynomial of degree 1) and do the same exploration as in 1 above and 2 above.

    What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

    What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?



More references and links to polynomial functions

Derivatives of Polynomial Functions.
Polynomial Functions, Zeros, Factors and Intercepts
Find Zeros of Polynomial Functions - Problems
Multiplicity of Zeros and Graphs Polynomials.
Graphs of Polynomial Functions - Questions.