Section 4.1 Polynomial Functions

A polynomial function is a function of the form a n, a n-1,…, a 1, a 0 are real numbers n is a nonnegative integer D: {x|x å real numbers} Degree is the largest power of x

Example: Determine which of the following are polynomials. For those that are, state the degree.

A power function of degree n is a function of the form where a is a real number a = 0 n > 0 is an integer.

(1, 1)(-1, 1) (0, 0) Power Functions with Even Degree

Summary of Power Functions with Even Degree 1.) Symmetric with respect to the y-axis. 2.) D: {x|x is a real number} R: {x|x is a non negative real number} 3.) Graph (0, 0); (1, 1); and (-1, 1). 4.) As the exponent increases, the graph increases very rapidly as x increases, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

(1, 1) (-1, -1) (0, 0) Power Functions with Odd Degree

Summary of Power Functions with Odd Degree 1.) Symmetric with respect to the origin. 2.) D: {x|x is a real number} R: {x|x is a real number} 3.) Graph contains (0, 0); (1, 1); and (-1, -1). 4.) As the exponent increases, the graph becomes more vertical when x > 1 or x < -1, but for -1 < x < 1, the graphs tends to flatten out and lie closer to the x-axis.

Graph the following function using transformations. (0,0) (1,1) (0,0) (1, -2)

(1,0) (2,-2) (1, 4) (2, 2)

If r is a Zero of Even Multiplicity Graph crosses x-axis at r. If r is a Zero of Odd Multiplicity Graph touches x-axis at r.

For the polynomial  fxxxx()  (a) Find the x- and y-intercepts of the graph of f. The x intercepts (zeros) are (-1, 0), (5,0), and (-4,0) To find the y - intercept, evaluate f(0) So, the y-intercept is (0,-20)

For the polynomial  fxxxx()  b.) Determine whether the graph crosses or touches the x-axis at each x-intercept. x = -4 is a zero of multiplicity 1 (crosses the x-axis) x = -1 is a zero of multiplicity 2 (touches the x-axis) x = 5 is a zero of multiplicity 1 (crosses the x-axis) c.) Find the power function that the graph of f resembles for large values of x.

d.) Determine the maximum number of turning points on the graph of f. At most 3 turning points. e.) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis. On the interval Test number: x = -5 f (-5) = 160 Graph of f: Above x-axis Point on graph: (-5, 160) For the polynomial  fxxxx()  154 2

 fxxxx()  On the interval Test number: x = -2 f (-2) = -14 Graph of f: Below x-axis Point on graph: (-2, -14) On the interval Test number: x = 0 f (0) = -20 Graph of f: Below x-axis Point on graph: (0, -20)

For the polynomial  fxxxx()  On the interval Test number: x = 6 f (6) = 490 Graph of f: Above x-axis Point on graph: (6, 490) f.) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

(6, 490) (5, 0) (0, -20) (-1, 0) (-2, -14) (-4, 0) (-5, 160)

Sections 4.2 & 4.3 Rational Functions 28

A rational function is a function of the form p and q are polynomial functions q is not the zero polynomial. D: {x|x å real numbers & q(x) = 0}.

Find the domain of the following rational functions. All real numbers x except -6 and -2. All real numbers x except -4 and 4. All Real Numbers 30

Vertical Asymptotes. Domain gives vertical asymptotes Reduce rational function to lowest terms, to find vertical asymptote(s). The graph of a function will never intersect vertical asymptotes. Describes the behavior of the graph as x approaches some number c Range gives horizontal asymptotes The graph of a function may cross intersect horizontal asymptote(s). Describes the behavior of the graph as x approaches infinity or negative infinity (end behavior) 31

Example: Find the vertical asymptotes, if any, of the graph of each rational function. Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes Vertical asymptote: x = -4 32

(3,2) (1,0) (2,0) (0,1) In this example there is a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.

Examples of Horizontal Asymptotes y = L y = R(x) y x y = L y = R(x) y x

Examples of Vertical Asymptotes x = c y x y x

If an asymptote is neither horizontal nor vertical it is called oblique. y x Note: a graph may intersect it’s oblique asymptote. Describes end behavior. More on this in Section 3.4.

Recall that the graph of is (1,1) (-1,-1) 37

Graph the function using transformations (1,1) (-1,-1) (3,1) (1,-1) (2,0) (3,2) (1,0) (2,0) (0,1)

Consider the rational function 1. If n < m, then y = 0 is a horizontal asymptote 2. If n = m, then y = a n / b m is a horizontal asymptote 3. If n = m + 1, then y = ax + b is an oblique asymptote, found using long division. 4. If n > m + 1, neither a horizontal nor oblique asymptote exists. 39

Example: Find the horizontal or oblique asymptotes, if any, of the graph of Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3

Oblique asymptote: y = x + 6

To analyze the graph of a rational function: 1) Find the Domain. 2) Locate the intercepts, if any. 3) Test for Symmetry. If R(-x) = R(x), there is symmetry with respect to the y-axis. If - R(x) = R(-x), there is symmetry with respect to the origin. 4) Find the vertical asymptotes. 5) Locate the horizontal or oblique asymptotes. 6) Determine where the graph is above the x-axis and where the graph is below the x-axis. 7) Use all found information to graph the function. 42

Example: Analyze the graph of

a.) x-intercept when x + 1 = 0: (-1,0) b.) y-intercept when x = 0: y - intercept: (0, 2/3) c.) Test for Symmetry: No symmetry

d.) Vertical asymptote: x = -3 Since the function isn’t defined at x = 3, there is a hole at that point. e.) Horizontal asymptote: y = 2 f.) Divide the domain using the zeros and the vertical asymptotes. The intervals to test are:

Test at x = -4 R(-4) = 6 Above x-axis Point: (-4, 6) Test at x = -2 R(-2) = -2 Below x-axis Point: (-2, -2) Test at x = 1 R(1) = 1 Above x-axis Point: (1, 1) g.) Finally, graph the rational function R(x)

(-4, 6) (-2, -2) (-1, 0)(0, 2/3) (1, 1)(3, 4/3) y = 2 x = - 3