Unit 2 Polynomial and Rational Functions PreCalculus 2-R

Review Problems 1 f (x) = – 4x 2 + 48x – 50 g(x) = 7x 2 – 28x Find the maximum value of the function. f (x) = – 4x 2 + 48x – 50 f (6) = 94 Find the maximum value of the function. g(x) = 7x 2 – 28x f (2) = –28 Review Problems 1

Review Problems 2 f (x) = x 2 – 12x + 2 D = ( ), R = [–34, ) 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 Review Problems 2

Review Problems 3 R (x) = 192x – 0.4x 2 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 Review Problems 3

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

Review Problems 5 Sketch the graph of the function.

Review Problems 6 Sketch the graph of the function.

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

How many local maxima and minima does the polynomial have? y = x 4 – 4x 2 + 4 1 maximum and 2 minima How many local maxima and minima does the polynomial have? y = 0.2x 5 + 2x 4 – 11.67x 3 – 66x 2 + 160x + 5 2 maximum and 2 minima Review Problems 8

Review Problems 9 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). Review Problems 9

Review Problems 10 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 Review Problems 10

Evaluate P(3) for: 38 Evaluate P(2) for: 234 Review Problems 11

Review Problems 12 Evaluate for: Use the Factor Theorem to choose a factor of Review Problems 12

Find a polynomial of degree 5 and zeros of –6, –2, 0, 2, and 6 Find a polynomial of degree 3 that has zeros of 2, –4, and 4, and where the coefficient of x2 is 6. Review Problems 13

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

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 Review Problems 15

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

Review Problems 17 Find all rational zeros of the polynomial.

Review Problems 18 Find all rational zeros of the polynomial.

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

Evaluate the expression (9 + 14i) + (7 – 11i) and write the result in the form a + bi. Evaluate the expression 10(–2 + 14i) and write the result in the form a + bi. –20 + 140i Review Problems 20

Review Problems 21 Evaluate the expression and write the result in the form a + bi. 16 + 8i Evaluate the expression and write the result in the form a + bi. 2 – 15i Review Problems 21

Review Problems 22 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. 2i Review Problems 22

Review Problems 23 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 Review Problems 23

Review Problems 24 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. Review Problems 24

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. Review Problems 25

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) Review Problems 26

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 Review Problems 27

Determine the correct graph of the rational function Review Problems 28

Determine the correct graph of the rational function Review Problems 29

Find the slant asymptote of the function y = x + 4 Review Problems 30

Determine the correct graph of the rational function Review Problems 31

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 = 2 x = 0 y = 0 (2.7, 12.3) none Review Problems 32

Review Problems 33 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 Review Problems 33

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 Review Problems 34

Review Problems 35 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 Review Problems 35

Answers f (6) = 94 f (2) = –28 D = ( ), R = [–34, ) $23,040, 240 units 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

Answers 38 234 The quotient is ; the remainder is 5 The quotient is 0, 2 or 4 negative 16 + 8i 2 – 15i i 2i –18 x = 4 + 3i, x = 4 – 3i z = –4 + 2i, z = –4 – 2i Answers

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