Smooth rounded corner. Smooth rounded corner. Smooth rounded corner

Transcription

1 3.2 Graphs of Higher Degree Polynomial Functions Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n-1,,a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0 is called a polynomial function of x of degree n. The number a n, the coefficient of the variable to the highest power, is called the leading coefficient. Graphs of Polynomial Functions Smooth and Continuous Graphs Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with nosharp corners. By continuous, we mean that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These ideas are illustrated in the figure below. Smooth rounded corner Smooth rounded corner Smooth rounded corner End Behavior of the Graph of a Polynomial Function The end behavior of the graph of a function to the far left or the far right is called its end behavior. To determine the end behavior of the graph of a polynomial function, we use the leading coefficient of the polynomial. The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial function f(x)= a n x n + a n-1 x n-1 + a n-2 x n a 1 x + a 0 (a n ¹ 0) eventually rises or falls. In particular, 1

2 2

3 Multiplicity of a Zero In mathematics, a zero, also sometimes called a root, of a polynomial function ƒ(x) is a value x such that ƒ(x) vanishes at x, that is ƒ(x) = 0. The real zeros of a polynomial function may be found by factoring (where possible) or by finding where the graph touches the x-axis. The number of times a zero occurs is called its multiplicity. Examples: 1. Function f(x) = x 4 (x 3) 2 (x + 7) (x +1) 5 has 4 zeros. They are: x = 0 with multiplicity 4. x = 3 with multiplicity 2. x = -7 with multiplicity 1. x = -1 with multiplicity 5 3

4 4

5 Multiplicity and x-intercepts If r is a zero of even multiplicity, then the graph touchesthe x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. If a zero is even and greater than 2 or odd and greater than 1, graphs tend to flatten out at zeros with multiplicity greater than one. Examples: 1. Function f(x) = (x+2) 2 (x 1) has 2 zeros: x=-2 is a zero of multiplicity 2 (even) at x = - 2 the graph touches the x-axis and turns around. x=1 is a zero of multiplicity 1 (odd) at x = 1 the graph crosses the x-axis. The graph of f is shown below. 2. Function f(x) = x 4 has 1 zero, x = 0 with multiplicity 4 (even), then the graph will touch the x-axis and turn around. Moreover, since 4 > 2, the graph flattens at x = 0. The graph of f is shown below. The graph touches the x-axis and flattens at x = 0. 5

6 6

7 Graphing a Polynomial Function f(x)= a n x n + a n-1 x n-1 + a n-2 x n-2 +.+a 1 x + a 0 (a n 0) Steps for Graphing a Polynomial Function: 1. Use the Leading Coefficient Test to determine the graph's end behavior. 2. Find all real zeros of the function with their multiplicity. (They are also x-intercepts) If the function has a zero of odd multiplicity, the graph of the function crosses the x-axis at that x-value. If the function has a zero of even multiplicity, the graph of the function only touches the x-axis at that x- value. 3. Find the y-intercept. 4. Additional points. 5. Connect all x-intercepts, y-intercept, additional points with the correct end behaviors and the correct behaviors at each x-intercept with smooth to get the graph. Remark: The property of even or odd functions can be used to get more information to graph. If the function is even, its graph is symmetric about y-axis. If the function is odd, its graph is symmetric about origin. Example1: Graph each of the functions below: 1. f(x) = x 4 2x Step 1: Determine end behavior. Because the degree is even (n = 4) and the leading coefficient, 1, is positive, the graph rises to the left and the right: Step 2: Find the x-intercepts (zeros of the function) by setting f(x) = 0. 7

8 Thus, f has 2 zeros: x = 1 multiplicity 2 even graph touches the x-axis and turns around. x = 1 multiplicity 2 even graph touches the x-axis and turns around. Step 3: Find the y-intercept. Replacing x with 0 in f (x)=x 4 2x we obtain the y-intercept is (0, 1). Step 4: Additional points. If x = 2, f(2) = = 9 (2, 9) If x = -2, f(2) = (-2) 4 2 (-2) = 9 (-2, 9) Because the function is even, so its graph is symmetric about y-axis. 8

9 Example 2: 9

10 Example 3: 10

11 Example 4: Sketch the following polynomial functions: 2. f (x)= x 3 2x 2 + x - Degree: - Leading coefficient: - End behavior: - y-int: - x-int: (zeros with multiplicity) - Additional points: 11

12 3. f (x)= -(x 1) 2 (x + 2)(x +1) - Degree: - Leading coefficient: - End behavior: - y-int: - x-int: (zeros with multiplicity) - Additional points: 12

13 4. f(x) = x 3 + x 2 9x 9 - Degree: - Leading coefficient: - End behavior: - y-int: - x-int: (zeros with multiplicity) - Additional points: 13