Mathematics Plus www.geocities.com/mathematicsplus
Curve Sketching: Polynomials
The function
P(x) = a(x � x1)(x � x2) ... (x � xn),

is a polynomial of degree n with zeros x1, x2, ... xn.
(Note that these zeros are not necessarily distinct.)

In the graph of a polynomial, the x- intercepts are equal to the zeros of the polynomial. Furthermore:

  • If x = a is a simple zero then (x � a) is a factor of P(x) and the polynomial cuts the x- axis at x = a.

  • If x = a is a double zero then (x � a)2 is a factor of P(x) and the polynomial has a turning point touching on the x- axis at x = a.

  • If x = a is a triple zero then (x � a)3 is a factor of P(x) and the polynomial has a horizontal point of inflection on the x- axis at x = a. In other words, the graph cuts, and is tangential to, the x- axis at x = a.

To sketch the graph of a polynomial, the following steps may be followed:
  1. Find the y- intercept.

  2. Find the x- intercept(s).

  3. Join the intercepts with a smooth and continuous curve*.

*This method is unreliable if the polynomial passes through the origin. In this case, determine the coordinates of a point on the graph of the polynomial, say at x = 1, and join it to the intercepts with a smooth and continuous curve.


The following animation illustrates the method for sketching polynomials. In the example,

x = -2 is a simple zero, x = -1 is a triple zero, and x = 1 is a double zero. There is a y- intercept at y = 2
Back to Top 

Produced by Dr T
�2001
Hosted by www.Geocities.ws

1