PreCalculus 2-R Unit 2 Polynomial and Rational Functions

Review Problems Find the maximum value of the function. f (x) = – 4x x – 50 f (6) = 94 Find the maximum value of the function. g(x) = 7x 2 – 28x f (2) = –28 1

Review Problems Find the domain and range of the function. f (x) = x 2 – 12x + 2 D = ( ), R = [–34, ) If a ball is thrown directly upward with a velocity of 80 ft/s, its height (in feet) after t seconds is given by y = 80t – 16t 2. What is the maximum height attained by the ball? 100 feet 2

Review Problems A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R (x) = 192x – 0.4x 2 where the revenue R (x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? $23,040, 240 units 3

Review Problems Sketch the graph of the function. P(x) = (x – 3)(x + 4) Sketch the graph of the function. 4

Review Problems Sketch the graph of the function. 5

Review Problems Sketch the graph of the function. 6

Review Problems Find the coordinates of all local extrema of the function. x = 2, y = –10, and x = –2, y = 22 y = x 3 – 12x + 6 Find the coordinates of all local extrema of the function. x = 0, y = 0, and x = 3, y = –27 y = x 4 – 4x 3 7

Review Problems How many local maxima and minima does the polynomial have? y = x 4 – 4x maximum and 2 minima How many local maxima and minima does the polynomial have? y = 0.2x 5 + 2x 4 – 11.67x 3 – 66x x maximum and 2 minima 8

Review Problems Find the quotient and remainder using division. The quotient is (x + 1); the remainder is –38. Find the quotient and remainder using division. The quotient is 1; the remainder is (2x + 3). 9

Review Problems Find the quotient and remainder using division. The quotient is ; the remainder is 5 Find the quotient and remainder using division. The quotient is the remainder is 5 10

Review Problems Evaluate P(3) for: 38 Evaluate P(2) for:

Review Problems 12 Find a polynomial of degree 3 that has zeros of 2, –4, and 4, and where the coefficient of x 2 is 6.

Review Problems Find a polynomial of degree 5 and zeros of –6, –2, 0, 2, and 6 13

Review Problems Find the polynomial of degree 4 whose graph is shown. 14

Review Problems List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. 0, 2 or 4 negative 15

Review Problems A polynomial P is given. Find all the real zeros of P. Sketch the graph of P. 16

Review Problems Find all rational zeros of the polynomial. 17

Review Problems Find all rational zeros of the polynomial. 18

Review Problems Find all of the real zeros of the polynomial and sketch the graph of P 19

Review Problems Evaluate the expression (9 + 14i) + (7 – 11i) and write the result in the form a + bi i Evaluate the expression 10(–2 + 14i) and write the result in the form a + bi. – i 20

Review Problems Evaluate the expression and write the result in the form a + bi i Evaluate the expression and write the result in the form a + bi. 2 – 15i 21

Review Problems Evaluate the expression i 17 and write the result in the form a + bi. i Evaluate the expression and write the result in the form a + bi. 2i2i 22

Review Problems Evaluate the expression and write the result in the form a + bi. –18 Find all solutions of the equation x 2 – 8 x + 25 = 0 and express them in the form a + bi. x = 4 + 3i, x = 4 – 3i 23

Review Problems Find all solutions of the equation and express them in the form a + bi. z = –4 + 2i, z = –4 – 2i A polynomial P is given. Factor P completely. 24

Review Problems Find the polynomial P(x) of degree 3 with integer coefficients, and zeros 4 and 3i. Factor the polynomial completely and find all its zeros. 25

Review Problems Factor the polynomial completely into linear factors with complex coefficients Find the x- and y-intercepts of the rational function x-intercept (6, 0), y-intercept (0, –1) 26

Review Problems Find the x- and y-intercepts of the rational function x-intercept (–3, 0), y-intercept (0, 3) Find the horizontal and vertical asymptotes of the rational function horizontal asymptote y = 0; vertical asymptote x = –8 27

Review Problems Determine the correct graph of the rational function 28

Review Problems Determine the correct graph of the rational function 29

Review Problems Find the slant asymptote of the function y = x

Review Problems Determine the correct graph of the rational function 31

Review Problems Use a Graphing Calculator to find vertical asymptotes, x- and y- intercepts, and local extrema, correct to the nearest decimal. (a) Find all vertical asymptotes. (b) Find x-intercept(s). (c) Find y-intercept(s). (d) Find the local minimum. (e) Find the local maximum. x = 0 y = 0 (2.7, 12.3) none x = 2 32

Review Problems Find the intercepts and asymptotes (a) Determine the x-intercept(s). (b) Determine the y-intercept(s). (c) Determine the vertical asymptote(s). (d) Determine the horizontal asymptote(s). no solution y = –3 x = –1, x = 3 y = 3 33

Review Problems Find all horizontal and vertical asymptotes (if any). (a) Find all horizontal asymptotes (if any). (b) Find all vertical asymptotes (if any). no solution x = 2, x = –2 34

Review Problems Find the intercepts and asymptotes (a) Determine the x-intercept(s). (b) Determine the y-intercept(s). (c) Determine the vertical asymptote(s). (d) Determine the horizontal asymptote(s). x = 2 y = –1 x = –8 y = 4 35

Answers f (6) = 94 f (2) = –28 D = ( ), R = [–34, ) 100 feet $23,040, 240 units x = 2, y = –10, and x = –2, y = 22 x = 0, y = 0, and x = 3, y = –27 1 maximum and 2 minima 2 maximum and 2 minima The quotient is (x + 1); the remainder is –38. The quotient is 1; the remainder is (2x + 3).

Answers The quotient is ; the remainder is 5 The quotient is the remainder is , 2 or 4 negative i – i i 2 – 15i i 2i2i –18 x = 4 + 3i, x = 4 – 3i z = –4 + 2i, z = –4 – 2i

Answers a 29. c c x-intercept (6, 0), y-intercept (0, –1) x-intercept (–3, 0), y-intercept (0, 3) horizontal asymptote y = 0; vertical asymptote x = –8 y = x + 4 x = 0 y = 0 (2.7, 12.3) none x = 2 no solution y = –3 x = –1, x = 3 y = 3 no solution x = 2, x = –2 x = 2 y = –1 x = –8 y = 4