M Polynomial Functions 1

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1 M Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n , be real numbers, with a n 0. The function defined by of degree n. The number leading coefficient. f ( x) a x n,..., a x a x a a n n is called a polynomial function of x, the coefficient of the variable to the highest power, is called the Note: The variable is only raised to positive integer powers no negative or fractional exponents. However, the coefficients may be any real numbers, including fractions or irrational numbers like or 7. Example 1: Given the following polynomial functions, state the leading term, the degree of the polynomial and the leading coefficient. a. P x x x x x 4 ( ) Leading Term: Degree: Leading Coefficient: b. P( x) (3x 2)( x 7) ( x 2) Leading Term: Degree: Leading Coefficient: 2 3 Graph Properties of Polynomial Functions Let P be any nth degree polynomial function with real coefficients. The graph of P has the following properties. 1. P is continuous for all real numbers, so there are no breaks, holes, jumps in the graph. 2. The graph of P is a smooth curve with rounded corners and no sharp corners. 3. The graph of P has at most n x-intercepts. 4. The graph of P has at most n 1 turning points. 5. The "middle" of the graph has hills and valleys, on either end the graph goes up or down "off the chart" and continues to go up (or down) for all real numbers off the scale on that side of the x-axis.

2 M Polynomial Functions 2 f ( x) x( x 2) ( x 1) Key Features of the Graph 1. y-intercept - the point where the graph crosses the y-axis (i.e. where x=0) 2. The x-intercepts - also called zeroes - of the graph are points where the graph crosses the x- axis. On the x-axis the y-value is 0 so these are x's for which f(x)=0. 3. What does the graph "look like" at each x-intercept? Take a magnifying glass and focus in on just that area, what function does it look like? 4. The End Behavior - at each side of the graph (left or right) the graph line goes "off the chart" meaning the function values are too big (+) or too small (-) for the scale. Which way does it go on each side. With the 4 key features and a little "connect the dots" logic (the graph has to be smooth and connected), we can create a fairly good sketch of the graph of a polynomial with plotting many points. That is the object of this section. HOW TO FIND THE KEY FEATURES 1. The y-intercept - Set x = 0 and find the function value. 2. The x-intercepts (ZEROES) If f is a polynomial and c is a real number for which root of f. If c is a zero of f, then c is an x-intercept of the graph of f. ( x c ) is a factor of f. f( c) 0 So if we have a polynomial in factored form, we know all of its x-intercepts. every factor gives us an x-intercept. every x-intercept gives us a factor., then c is called a zero of f, or a

3 M Polynomial Functions 3 Set the function = 0 and solve for x This usually means FACTORING!!!!!! If a polynomial is in factored form: set each factor to zero and solve, there are the zeroes. Example 2: Find the y-intercept and x-intercepts of the following functions P( x) x 5x 6x Example 3: f ( x) 3 x( x 3) (5x 2)(2x 1) (4 x) Behavior at Intercepts: Near an x-intercept, c, the shape of the function is determined by the factor (x-c) and the power to which it is raised. Power Functions: Even-degree power functions: f ( x) x 4 Odd-degree power functions: 5 f ( x) x

4 M Polynomial Functions 4 Let s look again at the graph on the first page f ( x) x( x 2) ( x 1) Notice the shape of the graph as it crosses the x-axis at each intercept. Definition: The multiplicity of a factor in a polynomial function is the power to which it is raised in the fully factored form of the polynomial. The multiplicity of the factor (x-c) with no higher power is 1. Compare the multiplicities of the factors in the above polynomial with the shape at each corresponding intercept. The graph of looks like the function near the x-intercept c. Since we know the shape of a power function, we have the following rules: P( x) ( x c) k f ( x) x k Even multiplicity: touches x-axis, but doesn t cross (looks like a parabola there). Odd multiplicity of 1: crosses the x-axis (looks like a line there). Odd multiplicity 3 : crosses the x-axis and looks like a cubic there. Example 4: Find the zeroes of the following polynomials and state the graph behavior at each x- intercept. 4 ( ) 2 1 P x x x x and f x x x x 5 8 ( ) 3( 4) ( 9) ( 1)

5 M Polynomial Functions 5 4. End Behavior of a Polynomial The end behavior of a polynomial is determined by its degree and the sign (positive or negative) of its leading coefficient. Remember that the degree is the highest power of x in the fully multiplied our form of a polynomial. If a polynomial is given in factored form, to get the degree, add the multiplicities of all the factors: If a polynomial is given in factored form, to find the leading coefficient, multiply the first number on the outside of all the factors times the coefficient on the x-term of each factor raised to the power that is its multiplicity. For example, p( x) 3 x ( x 1) (3x 5) 6 x, the leading coefficient is 2 3(1) (3)( 1) 9 Degree: is it odd or even? Sign: is the leading coefficient positive or negative? Determine the end behavior with the following rules: Odd Degree Odd Degree Even Degree Even Degree Sign (+) Sign (-) Sign (+) Sign (-) Find the end behavior for P( x) x( x 2)( x 4) 3 f ( x) ( x 3)( x 5)( x 5)( x 9) 4 g( x) ( x 3) ( x 4) 2 2

6 M Polynomial Functions 6 Steps to graphing other polynomials: 1. Find the y-intercept by finding P(0). 2. Factor and find x-intercepts. 3. Mark x-intercepts on x-axis. 4. For each x-intercept, determine the behavior. Even multiplicity: touches x-axis, but doesn t cross (looks like a parabola there). Odd multiplicity of 1: crosses the x-axis (looks like a line there). Odd multiplicity 3 : crosses the x-axis and looks like a cubic there. Note: It helps to make a table as shown in the examples below. 5. Determine the leading term. Degree: is it odd or even? Sign: is the coefficient positive or negative? 6. Determine the end behavior. Which direction does the graph go at each end? 7. Draw the graph, being careful to make a nice smooth curve with no sharp corners. Note: without calculus or plotting lots of points, we don t have enough information to know how high or how low the turning points are. Example 5: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if possible and have correct end behavior. 2 P x x x x 3 ( ) 2 3

7 M Polynomial Functions 7 Example 6: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if possible and have correct end behavior. 2 P( x) x 2 ( x 4) Example 7: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if possible and have correct end behavior. P x x x x ( )

8 M Polynomial Functions 8 Example 8: Given the graph of a polynomial determine what the equation of that polynomial. Example 8: Given the graph of a polynomial determine what the equation of that polynomial.

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