Section 2.2 Polynomial Functions of Higher Degree

A polynomial function is a continuous graph with smooth turns and curves that can be written in the form f(x) = a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 1 x 1 + a 0 where a n is the leading coefficient of the polynomial function and a 0 is the constant of the polynomial function. In factored form a polynomial function is written in the form f(x) = a n (x − c) n or f(x) = a n (x − c) n (x − d) k (x − b) r … etc

On your graphing calculator graph in Y 1 the following. a n = 2 a 0 = 8 Make your window [-5, 5] and [-5, 20].

On your graphing calculator graph Y 2 the following. a n = 2 a 0 = (-2) 2 (1) 3 = 8

The simplest form of a polynomial function is called a power function and is in the form y = x n The power functions with an even exponent are tangent to the x-axis. Example: y = x 2 If n is odd and n > 1, then the graph of y = x n has a change in concavity at the x-axis. Example: y = x 3

Example 1 Sketch the graph by hand using your knowledge of transformations and power functions. A.f (x) = −(x + 2) 2 B.f (x) = (x − 3) 3 + 4

A.f(x) = −(x + 2) 2

B.f(x) = (x − 3) 3 + 4

Leading Coefficient Test As x moves without bound to the left or to the right, the graph of the polynomial function f(x) = a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 1 x 1 + a 0 eventually rises or falls in the following manner.

1.The degree is odd and the leading coefficient is positive: Left hand behavior: falls Right hand behavior: rises 2.The degree is odd and the leading coefficient is negative: Left hand behavior: rises Right hand behavior: falls

3.The degree is even and the leading coefficient is positive: Left hand behavior: rises Right hand behavior: rises 4.The degree is even and the leading coefficient is negative: Left hand behavior: falls Right hand behavior: falls

Rises for end behavior means as x → , f (x) →  as x → − , f (x) →  Falls for end behavior means as x → − , f (x) → −  as x → , f (x) → − 

Example 2 Describe the left-hand and right-hand behavior (end behaviors) of the graph of each function. a.f(x) = -x 4 + 7x 3 – 14x – 9 falls on the left and right b.g(x) = 5x 5 + 2x 3 – 14x falls on the left and rises on the right

Zeros of Polynomial Functions For a polynomial function f of degree n, the following is true. 1.The function f has, at most, n real zeros. 2.The graph of f has, at most, n – 1 turning points Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versus.

If f is a polynomial function and a is a real number, the following statements are equivalent. 1.x = a is a zero of the function f. 2.x = a is a solution of the polynomial equation f (x) = 0. 3.(x – a) is a factor of the polynomial f (x). 4.(a, 0) is an x-intercept of the graph of f.

Example 3 Find the x-intercepts of the graph of f (x) = x 3 – x 2 – x Set the function equal to 0. x 3 – x 2 – x + 1 = 0

2. Factor by grouping. x 3 – x 2 – x + 1 = 0 (x 3 – x 2 ) + (– x + 1) = 0 x 2 (x – 1) – 1(x – 1) = 0 (x – 1) (x 2 – 1) = 0 (x – 1)(x + 1) (x – 1) = 0 (x – 1) 2 (x + 1) = 0 3.Set each different factor equal to 0 and solve for x. x – 1 = 0x + 1 = 0 x = 1x = -1 The x-intercepts are (1, 0) and (-1, 0).

Note that in the last example, 1 is a repeated zero. In general, a factor (x – a) k, k > 1, yields a repeated zero x = a of multiplicity k. If k is odd, the graph has a change of concavity (cc) on the x-axis, at x = a. If k is even, the graph is tangent (t) to the x-axis at x = a. If k = 1, the graph crosses (s) the x-axis at x = a. Graph the last example on your calculator.

f(x) = x 3 – x 2 – x + 1 = (x – 1) 2 (x + 1)

Example 4 Find all the real zeros for the following. State the multiplicity of each zero. Describe the behavior of the graph around the zeros. Determine the number of turning points for each graph. A.f (x) = x 3 – 12x x B.f (x) = -2x 5 + 2x 3

A.f (x) = x 3 – 12x x The behavior around x = 0 is the graph crosses at (0, 0). The behavior around x = 6 is the graph is tangent at (6, 0). There are at most 2 turning points on the graph.

B.f(x) = -2x 5 + 2x 3 The behavior around x = 0 is the graph is a change in concavity at (0, 0). The behavior around x = 1 and x = -1 is the graph crosses at (1, 0) and (-1, 0). There are at most 4 turning points on the graph.

Example 5 Find a polynomial function that has the given zeros x = 4, -3, 3, 0 x = 0, x = 4, x = -3, x = 3

Example 6 Find a polynomial function of degree n that has the given zeros x = -5, 1, 2 and n = 4 The degree of the factors must add up to 4

Graphing Polynomial Functions 1.Apply the Leading Coefficient Test. 2.Find the zeros of the polynomial.

To graph a polynomial function, you can use the fact that the function can change signs only at its zeros. Between two consecutive zeros, the polynomial must be either entirely positive or entirely negative.

Example 7 Sketch the graph f (x) = x 3 – 2x 2 1.Find the end behaviors of the function. The graph falls on the left and rises on the right. 2.Find the zeroes of the function. 3.Now sketch the graph. Write s, t, or cc by the zeros.

t s

When looking at these graphs of polynomial functions you should be able to estimate the intervals at which the graphs are increasing or decreasing.