Section 5.1 Polynomial Functions & Models Polynomial Function

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1 Week 8 Handout MAC 1105 Professor Niraj Wagh J Section 5.1 Polynomial Functions & Models Polynomial Function A polynomial function is of the form: f (x) = a n x n + a n 1 x n a 1 x 1 + a 0 where a n, a n-1, a 1, a 0 are real numbers and n is a nonnegative integer. The domain of a polynomial function is the set of all real numbers. Example: P(x) = 5x 4 + 3x 3 + 2x 2 + 3x +8 The leading term is 5x 4 since it has the highest power of n. The leading coefficient is 5. The overall coefficients of the polynomial are 5, 3, 2, 3, and 8. The constant term (term with no x ) is 8. The overall degree of the polynomial is 4 since that s the highest degree. Zeros The possible zeros of a graph are the points where the graph hits the x-axis. That is when f(x) = 0. Multiplicity The multiplicity refers to the number of times a particular number is a zero of a graph. It is essentially the power of a factor. N. Wagh 1

2 Example: f (x) = (x 5) 2 (x 3) If we set f(x) = 0, we note the possible zeros are 5 and 3. 5 has a multiplicity of 2 (since there are 2 zeros of 5 ) and 3 has a multiplicity of 1 (since there is only one zero of 3 ). The polynomial has a total number of three zeros. Even/Odd Multiplicity If a zero has an EVEN multiplicity: è The graph of the function TOUCHES the x-axis at the zero. If a zero had an ODD multiplicity: è The graph of the function CROSSES the x-axis at the zero. Turning Points If f is a polynomial of degree n, then the graph of f has at most n-1 turning points. Example: y = x 4-2x 2 + x y = x 4-2x 3 Turning Points 1 Turning Point Both graphs have a degree of 4, but the graph on the left has 3 turning points at (-1, -2), (0.25, 0.25), & (1, 0) and the one on the right has one at (0.75, -1.25). N. Wagh 2

3 5.1 Examples Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. 1. Zeros: -2, 2, 3; degree 3 2. Zeros: -1 with multiplicity 1; 3 with multiplicity 2; degree 3 For the polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is find the power function that the graph of f resembles for large values of lxl. 3. f (x) = 4(x + 4)(x + 3) 3 N. Wagh 3

4 PRACTICE. 1. For the polynomial function: (a) List each real zero and its multiplicity (b) Determine whether the graph crosses or touches the x-axis at each x- intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is find the power function that the graph of f resembles for large values of x f (x) = x ( x 1) 3 2. Find the polynomial function with the given zeros. Zeros: -2 (multiplicity 2), 0 (multiplicity 1), 2 (multiplicity 1) Work on MML HW 5.1. If you have any questions, let me know. J N. Wagh 4

5 Sections 5.2 & 5.3 Rational Functions (Properties & Graphing) Rational Function A rational function is a ratio of two polynomials. The domain is the set of all real numbers except those for which the denominator equals 0 (which would make the function undefined). Asymptote An asymptote is a (dashed) line that a curve approaches as it heads toward positive or negative infinity. There are three types of asymptotes: vertical, slant/oblique, and horizontal. N. Wagh 5

6 Vertical Asymptotes The line x = a is a vertical asymptote (V.A.) of the function y = f(x) if y approaches ± as x approaches a from the right or left. è To find the vertical asymptotes of a function, factor the denominator, set it equal to 0, and solve for x. Slant/Oblique Asymptotes A slant or oblique asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator. è To find the slant/oblique asymptote, we have to perform long division. The slant asymptote will be of the form y = mx+b. Horizontal Asymptotes The line y = b is a H.A. of the function y = f(x) if y approaches b as x approaches ±. They basically show the general behavior of the rational function at the ends of the graph. -> To find the horizontal asymptotes of a function, there are two cases: Case 1: The numerator has a LOWER leading term (lower power) than the denominator. In this case the horizontal asymptote is y = 0. Example: f (x) = x3 + 2x 2 + 5x + 3 x 4 +8x 2 + 7x + 9 Horizontal Asymptote: y = 0 N. Wagh 6

7 Case 2: The numerator has the SAME leading term (same power) as the denominator. In this case look at the leading COEFFICENTS and that is your horizontal asymptote. Example: f (x) = 2x 4 + 2x 2 + 5x + 3 5x 4 +8x 2 + 7x + 9 Horizontal Asymptote: y = 2 5 NOTE: You CAN cross horizontal/slant asymptotes (if you have to) but you CANNOT cross vertical asymptotes. Steps to Graph Rational Functions 1. Factor and find domain. 2. Find X & Y intercepts. To find the x-intercepts, let y = 0. Conversely, to find the y-intercepts, let x = Find the horizontal asymptote(s). 4. Find the vertical asymptote(s). Do this by setting the denominator of the rational function equal to 0, and solve for X. 5. Determine the behavior of the graph as it approaches the vertical asymptote. It s helpful to create a sign chart. 6. Sketch the graph with what you know! N. Wagh 7

8 5.2 & 5.3 Examples Graph the following rational functions using the steps above. 1. R(x) = 3x + 5 x 6 N. Wagh 8

9 PRACTICE. PRACTICE. PRACTICE. R(x) = x x ( + x 2 All done with 5.2 & 5.3! Work on MML HW 5.2 & 5.3. If you have any questions, let me know. J N. Wagh 9