Section 4.1 Polynomial Functions A polynomial function is a function of the form f ( x ) a n x n a n 1 x n 1 a 1 x a 0 an , an-1 ,…, a1 , a0 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) f ( x ) 3 x 2 4 x 5 Polynomial of degree 2 (b) h ( x ) 3 x 5 Not a polynomial 5 3x (c) F ( x ) 5 2x Not a polynomial A power function of degree n is a function of the form f ( x) a n x where a is a real number a =0 n > 0 is an integer. n Power Functions with Even Degree 10 yx 8 8 yx 4 6 4 2 (-1, 1) 2 (1, 1) 1 (0, 0) 0 1 2 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. Power Functions with Odd Degree 10 yx 9 6 2 (-1, -1) 1 yx 5 2 (0, 0) 2 0 6 10 (1, 1) 1 2 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. f ( x ) 4 2 x 1 2( x 1) 4 4 4 15 15 (1,1) 5 (0,0) 0 5 15 yx (0,0) 5 0 (1, -2) 15 4 y 2 x 4 5 15 15 (1, 4) (1,0) 5 0 (2, 2) 5 (2,-2) 5 0 5 15 15 y 2 x 1 4 y 2x 1 4 4 If r is a Zero of Even Multiplicity Graph touches x-axis at r. If r is a Zero of Odd Multiplicity Graph crosses x-axis at r. For the polynomial 2 f ( x ) x 1 x 5 x 4 (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) f (0) (0 1)(0 5)(0 4) 20 So, the y-intercept is (0,-20) For the polynomial 2 f ( x ) x 1 x 5 x 4 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. f (x) x 4 For the polynomial 2 f ( x ) x 1 x 5 x 4 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 x 4 Test number: x = -5 f (-5) = 160 Graph of f: Above x-axis Point on graph: (-5, 160) For the polynomial 2 f ( x ) x 1 x 5 x 4 On the interval 4 x 1 Test number: x = -2 f (-2) = -14 Graph of f: Below x-axis Point on graph: (-2, -14) On the interval 1 x 5 Test number: f (0) = -20 Graph of f: x= 0 Below x-axis Point on graph: (0, -20) For the polynomial 2 f ( x ) x 1 x 5 x 4 On the interval 5 x 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. 500 (6, 490) 300 (-1, 0) (-5, 160) 100 8 (-4, 0) 6 4 (0, -20) 2 0 2 100 (-2, -14) 300 4 (5, 0) 6 8 Sections 4.2 & 4.3 Rational Functions 28 A rational function is a function of the form p( x) R( x ) q( x) • 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. x 1 x 1 (a) R ( x ) 2 x 8 x 12 x 6 x 2 All real numbers x except -6 and -2. x4 x4 (b) R ( x ) 2 x 4 x 4 x 16 All real numbers x except -4 and 4. 5 (c) R ( x ) 2 x 9 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. 3 3 (a) R ( x ) 2 x 1 ( x 1)( x 1) Vertical asymptotes: x = -1 and x = 1 x 5 (b) R ( x ) 2 x 1 No vertical asymptotes 1 x3 x3 (c) R ( x ) 2 x4 x x 12 ( x 3)( x 4) Vertical asymptote: x = -4 32 (3,2) (0,1) (2,0) (1,0) 1 f (x) 1 x2 In this example there is a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. lim R( x) L x Examples of Horizontal Asymptotes y y = R(x) y=L x lim R( x) L x y y=L x y = R(x) Examples of Vertical Asymptotes x=c y x=c y x 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. 1 Recall that the graph of f ( x) x is (1,1) (-1,-1) 37 1 Graph the function f ( x ) x 2 1 using transformations (1,1) (3,1) (2,0) (1,-1) (-1,-1) f ( x) 1 x 1 f ( x) x2 (3,2) (0,1) (2,0) (1,0) Consider the rational function p ( x ) an x n an 1 x n 1 a1 x a0 R( x) q ( x ) bm x m bm 1 x m 1 b1 x b0 1. If n < m, then y = 0 is a horizontal asymptote 2. If n = m, then y = an / bm 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 3x 4 x 15 (a) R ( x ) 3 2 x 4x 7x 1 2 Horizontal asymptote: y = 0 2 x2 4 x 1 (b) R ( x ) 3x 2 x 5 Horizontal asymptote: y = 2/3 x 4x 1 (c) R ( x ) x2 2 x6 x 2 x2 4 x 1 - x 2 2 x 6x 1 - 6 x 12 13 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 2x2 4x 6 Example: Analyze the graph of R ( x ) x2 9 2 x 2 x 3 R( x) x 3 x 3 2 2 x 3 x 1 x 3 x 3 2 x 1 , x3 x 3 Domain: x x 3, x 3 2 x 1 R( x) x 3 a.) x-intercept when x + 1 = 0: (-1,0) 2 ( 0 1) 2 b.) y-intercept when x = 0: R ( 0 ) ( 0 3) 3 y - intercept: (0, 2/3) c.) Test for Symmetry: 2( x 1) R( x) ( x 3) R(x) R(x) R(x) No symmetry 2 x 1 R( x) , x3 x 3 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: x 3 3 x 1 1 x x 3 3 x 1 1 x Test at x = -4 Test at x = -2 Test at x = 1 R(-4) = 6 R(-2) = -2 R(1) = 1 Above x-axis Below x-axis Above x-axis Point: (-4, 6) Point: (-2, -2) Point: (1, 1) g.) Finally, graph the rational function R(x) x=-3 10 (-4, 6) 5 (1, 1) (3, 4/3) y=2 8 6 (-2, -2) 4 2 0 5 10 2 (-1, 0) 4 (0, 2/3) 6