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Math 1330 Section 2. Polynomial Functions
Our objectives in working with polynomial functions will be, first, to gather information about the graph of the function and, second, to use that information to generate a reasonably good graph without plotting a lot of points. In later examples, we’ll use information given to us about the graph of a function to write its equation.
A polynomial function is a function of the form
1 0
2 2
2 2
1 P ( x ) ax a 1 x a x ax ax a n n
n n
n = (^) n + + + + + + − −
− − ⋯
where an ≠ 0 , a (^) 0 , a 1 ,…, an are real numbers and n is a whole number.
The number an is called the leading coefficient. The degree of the polynomial function is n.
P ( 0 )= a 0.
From college algebra, you should be familiar with the graphs of f ( x )= x^2 and g ( x )= x^3.
The graph of f ( x )= xn^ , n > 0 , n is even, will resemble the graph of f ( x )= x^2 , and the graph of
f ( x )= xn^ , n > 0 , n is odd, will resemble the graph of f ( x )= x^3.
Next, you will need to be able to describe the end behavior of a function.
If the degree of the function is even and an > 0 , then the end behavior of the function is
If the degree of the function is even and an < 0 , then the end behavior of the function is
If the degree of the function is odd and an > 0 , then the end behavior of the function is
Example 1: Find the x and y intercepts of the graph of the function. State the degree of the
function. P ( x )= ( x − 3 )( x + 1 )( x + 2 ).
Example 2: Find the x and y intercepts of the graph of the function. State the degree of the
function. P ( x )= x^3 − x^2 − x + 1.
Example 3: State the x and y intercepts of the graph of the function P ( x )= x ( x − 3 )( x + 1 ). State
the degree of the function. Then sketch the graph of the function, labeling all intercepts. Show the correct behavior through each x intercept, and show the proper end behavior.
Example 4: State the x and y intercepts of the graph of the function ( )( )( )
2 P ( x )= 3 − x x + 1 x + 5. State the degree of the function. Then sketch the graph of the function, labeling all intercepts. Show the correct behavior through each x intercept, and show the proper end behavior.
Example 5: State the x and y intercepts of the graph of the function
P ( x )= 4 ( x + 1 ) 2 ( x − 2 )^2 ( 1 − x ). State the degree of the function. Then sketch the graph of the
function, labeling all intercepts. Show the correct behavior through each x intercept, and show the proper end behavior.
With some problems, you can use transformations to graph polynomial functions.
Example 9: Graph using transformations: f ( x )= x^3 − 2
Example 10: Graph using transformations: f ( x )= −( x + 1 )^4 + 3
