Polynomial Graph

    The graphs of linear polynomials and quadratic polynomials are lines and parabolas, respectively. The emphasis of this topic is on a polynomial graph of a polynomial function whose degree is greater than 2.

Theorem (Polynomial Graph) Properties. Suppose is a polynomial function of degree

    (i) The polynomial graph of is far from the -axis when is large.
    
    (ii) The polynomial graph of crosses the -axis at most times.
    
    (iii) The polynomial graph of has at most turning points (relative extrema).
    
    (iv) The polynomial graph represents a function whose domain is all real numbers.
    
    (v) The polynomial graph represents a continuous function.
    

Example (Polynomial Graph) Properties. The graph of the fourth degree polynomial defined by



is shown below: Notice that when grows larger and larger the values of also grow larger and larger. We write this using as Similarly, as Also notice that the graph of crosses the -axis 3 times. Notice that the graph of has 3 turning points.
    

    The following theorem is known as the intermediate value theorem for polynomials and basically says that if is a continuous function on the closed interval and is some number strictly between and , then there exists at least one number between and such that

Theorem (Polynomial Graph) Intermediate Value Theorem. If is a polynomial function and for then takes on every value between and in the interval

    The intermediate value theorem says that if and have different signs then the function has at least one between and with Or plainly said, if and have different signs then the is an -intercept between and This is an important existence theorem because solving the equation may be difficult, and even algebraically impossible.  

Example (Polynomial Graph) Intermediate Value Theorem. Show that the polynomial function defined by



has an -intercept between and .

    Solution. Can we use the intermediate value theorem? We check that and have different signs which is true because and Therefore, the intermediate value theorem applies and so the function must have at least one -intercept between and As a side note, since and does this mean that does not have any -intercepts between and The answer is no, notice that and The graph of follows:

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