Review Exercises for Chapter 2
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1 Review Eercises for Chapter 7 Review Eercises for Chapter. (a) Vertical stretch Vertical stretch and a reflection in the -ais Vertical shift two units upward (a) Horizontal shift two units to the left. (a) Vertical shift four units downward Reflection in the -ais and a vertical shift four units upward Horizontal shift three units to the right Vertical shrink (each -value is multiplied b and a vertical shift one unit downward,
2 Chapter Polnomial and Rational Functions. g. f Verte:, 7 Verte:, Ais of smmetr: -intercepts:,,, Ais of smmetr: -intercepts:,,,. f. h Verte:, Ais of smmetr: ± ± -intercepts: ±, Verte: 7 7, 7 Ais of smmetr: ± ± ± 7 -intercepts: ± 7, 7. ft t t. f t t t t Verte:, Verte:, Ais of smmetr: t t t t Ais of smmetr: -intercepts:,,, t ± t ± t-intercepts: ±,
3 Review Eercises for Chapter. h. f Verte:, Ais of smmetr: Verte:, Ais of smmetr: ± ± ± -intercepts: ±, No real zeros -intercepts: none. h. f Verte:, Verte:, Ais of smmetr: Ais of smmetr: B the Quadratic Formula, -intercepts: ±, ±. B the Quadratic Formula, The equation has no real zeros. -intercepts: None ± i ± i.. f Verte:, Ais of smmetr: B the Quadratic Formula, -intercepts: ±, ±.
4 Chapter Polnomial and Rational Functions. f Verte:, Ais of smmetr: ± -intercepts: ±, ± ±. Verte:, f a Point:, a a a Thus, f.. Verte:, f a Point:, a a a f a 7. Verte:, f a Point:, a a Thus, f.. Verte:, f a Point:, a f a a a. (a) Area Area The maimum area occurs at the verte when and. The dimensions with the maimum area are meters and meters.. R p p (a) R $, R $,7 R $, The maimum revenue occurs at the verte of the parabola. b $ a R $, The revenue is maimum when the price is $ per unit. The maimum revenue is $,.
5 Review Eercises for Chapter. C 7,.. The minimum cost occurs at the verte of the parabola. Verte: b units a. Approimatel units should be produced each da to ield a minimum cost ± ,. The age of the bride is approimatel ears when the age of the groom is ears. Age of groom 7 Age of bride., f., f., f 7 Transformation: Reflection in the -ais and a horizontal shift four units to the right f is a reflection in the -ais and a vertical stretch of the graph of. Transformation: Reflection in the -ais and a vertical shift two units upward., f 7., f., f 7 f is a shift to the right two units and a vertical stretch of the graph of. Transformation: Horizontal shift three units to the right f is a vertical shrink and a vertical shift three units upward of the graph of.. f. The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. f The degree is odd and the leading coefficient is positive. The graph falls to the left and rises to the right.. g. h 7 The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right. The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right.
6 Chapter Polnomial and Rational Functions. f. f 7 Zeros:, 7, all of multiplicit (odd multiplicit) Turning points: Zeros: of multiplicit (odd multiplicit) of multiplicit (even multiplicit) Turning points:. f t t t. t t tt Zeros: t, ± all of multiplicit (odd multiplicit) Turning points: f Zeros: of multiplicit (even multiplicit) of multiplicit (odd multiplicit) Turning points: 7. f. g Zeros: of multiplicit (even multiplicit) Zeros: of multiplicit (even multiplicit) of multiplicit (odd multiplicit) Turning points: of multiplicit (odd multiplicit) of multiplicit (odd multiplicit) Turning points:. f. (a) The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. Zero: (, ) f g (a) The degree is odd and the leading coefficient,, is positive. The graph rises to the right and falls to the left. g The zeros are and. g (, ) (, )
7 Review Eercises for Chapter. f. h (a) The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right. Zeros:,, (a) The degree is even and the leading coefficient,, is negative. The graph falls to the left and falls to the right. g f 7 The zeros are,, and. (, ) (, ) (, ) h ( (, ) (, (, (. (a) f. (a) f... f 7 7 f The zero is in the interval,. Zero:. f The onl zero is in the interval,. It is.7.. (a) f. (a) f 7 f f There are two zeros, one in the interval, and one in the interval, Zeros:.,.77 There are zeros in the intervals, and,. The are. and.. 7. Thus, ).. ) 7 7
8 Chapter Polnomial and Rational Functions. ) Thus,.. ). ) Thus,.. ). Thus, Thus,.. 7. f (a) Yes, is a zero of f. Yes, is a zero of f. Yes, is a zero of f. No, is not a zero of f.
9 Review Eercises for Chapter. f (a) Yes, is a zero of f. No, is not a zero of f. Yes, is a zero of f. No, is not a zero of f.. f (a) 7 Thus, f. f. gt t t t (a) 7 Thus, g 7. Thus, g.. f ; Factor:. f (a) The remaining factors of f are 7 and. Zeros: (e) Yes, is a factor of f. 7 7 f 7 7,, 7 (a) Yes, is a factor of f. The remaining factors are and. Zeros: (e) f,, 7
10 Chapter Polnomial and Rational Functions. f 7. Factors: (a) Both are factors since the remainders are zero. The remaining factors are and. Zeros: (e), f,,, 7 f (a) Yes, and are both factors of f. The remaining factors are and. Zeros: (e) f,,,. i. i 7. i i i. i i i. 7 i i 7 i i 7i 7. i i i i i i 7. i i i i i 7. i i i i i i 7 i 7. i i i i i 7. i i i i i i i i i i i i i 7. i i i i i i i i i i 7. i i i i i i i i 7 7i 7 7i i i
11 Review Eercises for Chapter i i i i i i i i i i i i i i i 7. i i i i i i i i i i i i i i i i i i i i i 7.. ± ± i ± i ± i.. ± ± i 7 b ± b ac a ± 7 ± ± i7 ± 7 i. f. f. f. f Zeros:, Zeros:, Zeros:, Zeros:, ±i 7. f i i. Zeros:,, i, i f i i Zeros:,, ± i. f. f Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ± ±, ±, ±, ±, Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ±
12 Chapter Polnomial and Rational Functions. f. Possible rational zeros: ±, ±, ±, ±, ±, ± The zeros of f are,, and. f 7 Possible rational zeros: So, f 7. Zeros:,, 7 ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±. f 7. Possible rational zeros: ±, ±, ±, ± 7 7 The zeros of f are and. f Possible rational zeros: ±, ±, ±, ±, ±, ± So, f. Zeros:,. f Possible rational zeros: ±, ±, ±, ±, ±, ± The real zeros of f are, and.. f Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ± ± ±, ±, ±, ±, ±, ±, ±, ±,, 7 So, f. Zeros:,, ±
13 Review Eercises for Chapter f i i 7. Since i is a zero, so is i. Multipl b to clear the fraction. 7 Note: f a 7, where a is an real nonzero number, has zeros,, and ±i.. Since i is a zero and the coefficients are real, i must also be a zero.. f, Zero: i Since i is a zero, so is i. f i i i i i 7 i i i i i i i f i i, Zeros: ± i,. h. g 7, Zero: i Since i is a zero, so is i. i i i i h i i Zeros: ± i, i i i i i i Since i is a zero, so is i i i i i i i g i i i i Zeros: ± i,, i i i i 7 i i i i. f. f One zero is. Since i is a zero, so is i. i i 7 i i i Zeros:,, i 7 i i f i i i i Zeros:,, i, i i i
14 Chapter Polnomial and Rational Functions. g 7 7 The zeros of are,. The zeros of g are,,. g. g, Zero:. g B the Quadratic Formula the zeros of are ± i. The zeros of g are of multiplicit, and ± i. g i i i i f B the Quadratic Formula, the zeros of 7 are ± 7 The zeros of f are,, i, i. f i i ± ± i. 7. g. h g has two variations in sign, so g has either two or no positive real zeros. g g has one variation in sign, so g has one negative real zero. h has three variations in sign, so h has either three or one positive real zeros. h h has two variations in sign, so h has either two or no negative real zeros.. f. g (a) Since the last row has all positive entries, is an upper bound. 7 Since the last row entries alternate in sign, is a lower bound. (a) 7 Since the last row has all positive entries, is an upper bound. Since the last row entries alternate in sign, is a lower bound.
15 Review Eercises for Chapter. f. f. f Domain: all real numbers ecept Domain: all real numbers ecept and Domain: all real numbers ecept. f. f. f Domain: all real numbers Vertical asmptote: Vertical asmptote: none Horizontal asmptote: Horizontal asmptote: 7. h, Vertical asmptote: Horizontal asmptote:. h Vertical asmptotes: Horizontal asmptotes: none,. f (a) Domain: all real numbers ecept No intercepts Vertical asmptote: Horizontal asmptote: ± ± ±. f (a) Domain: all real numbers ecept No intercepts Vertical asmptote: Horizontal asmptote:
16 Chapter Polnomial and Rational Functions. g. h (a) Domain: all real numbers ecept -intercept: -intercept: Vertical asmptote: Horizontal asmptote:,, (a) Domain: all real numbers ecept -intercept: -intercept: Vertical asmptote: Horizontal asmptote:,, (, ) (, ) (, ( (, ). p. f (a) Domain: all real numbers (a) Domain: all real numbers Intercept:, Intercept:, Horizontal asmptote: Horizontal asmptote: ± ± ± (, ) (, ). f (a) Domain: all real numbers Intercept:, Horizontal asmptote: (, )
17 Review Eercises for Chapter. h (a) Domain: all real numbers ecept -intercept:, Vertical asmptote: Horizontal asmptote: 7. f (a) Domain: all real numbers Intercept: Horizontal asmptote:, ± ± ± 7 7 (, ) (, ). (a) Domain: all real numbers ecept ± Intercept:,. f, Vertical asmptotes:, Horizontal asmptote: ± ± ± ± (, ) (a) Domain: all real numbers ecept and -intercept: -intercept: none Vertical asmptote: Horizontal asmptote: 7, (, (
18 Chapter Polnomial and Rational Functions. f 7, (a) Domain: all real numbers ecept ± -intercept:, -intercept:, Vertical asmptote:. (a) Domain: all real numbers Intercept: Slant asmptote: f, Horizontal asmptote: (, ) (, (. f (a) Domain: all real numbers ecept -intercept: Vertical asmptote:, Using long division, Slant asmptote: f. 7 7 (, )
19 Review Eercises for Chapter. f (a) Domain: all real numbers ecept, -intercepts:, and -intercept: Vertical asmptote: Slant asmptote:,,,. f (a) Domain: all real ecept or - intercepts: Vertical asmptote: -intercept:, Using long division, f Slant asmptote:,,,,. (, ( (, (, (, ) ( ( (, ) (, ). C C., <. Horizontal asmptote: C.. As increases, the average cost per unit approaches the horizontal asmptote, C. $.. C (a) When p p, p < When When p, C $7 p, C $ p 7, C 7 $ 7 million. million. million. As p, C. No, it is not possible.
20 Chapter Polnomial and Rational Functions 7. (a) in. Because the horizontal margins total inches, must be greater than inches. The domain is >. in. in. The area of print is, which is square inches. in. 7 Total area 7 7 The minimum area occurs when.77 inches, so inches. The least amount of paper used is for a page size of about. inches b. inches.. The limiting amount of CO uptake is determined b the horizontal asmptote,.7.. mgdm hr..7.., <. < < Critical numbers: Test intervals: <,,,,,,, Test: Is <? B testing an -value in each test interval in the inequalit, we see that the solution set is:,.. Critical numbers:, Test intervals:, >, <, > Solution interval:,, Critical numbers:, ± Test intervals:,,,,,,, Test: Is? B testing an -value in each test interval in the inequalit, we see that the solution set is:,,.
21 Review Eercises for Chapter 7. <. < Critical numbers: Test intervals:,, <, <, > Solution interval:,, ) Critical numbers:, ± Test intervals:,,,,,,, Test: Is? B testing an -value in each test interval in the inequalit, we see that the solution set is:,,. <. 7 Critical numbers: Test intervals:,, <, >, < Solution intervals:,, Critical numbers: Test intervals:,,,,,,, Test: Is,,? B testing an -value in each test interval in the inequalit, we see that the solution set is:,,. > > Critical numbers:, Test intervals:,,, < > Solution interval:,, > 7. r > r >. r >. r >. r >.%. P t t t t t t, t t. False. A fourth-degree polnomial can have at most four zeros and comple zeros occur in conjugate pairs.. False. (See Eercise.) The domain of f is the set of all real numbers. t t das
22 Chapter Polnomial and Rational Functions. The maimum (or minimum) value of a quadratic function is located at its graph s verte. To find the verte, either write the equation in standard form or use the formula b a, f b a. If the leading coefficient is positive, the verte is a minimum. If the leading coefficient is negative, the verte is a maimum.. Answers will var. Sample answer: Polnomials of degree n > with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. Setting the factors equal to zero and solving for the variable can find the zeros of a polnomial function. To solve an equation is to find all the values of the variable for which the equation is true.. An asmptote of a graph is a line to which the graph becomes arbitraril close as increases or decreases without bound. Problem Solving for Chapter. f a b c d a ak b ak bk c k) a b c d a ak ak b c ak b ak bk ak bk c d ak bk c ak bk ck ak bk ck d Thus, f a b c d ka ak b ak b c ak bk ck d and f k ak bk ck d. Since the remainder r ak bk ck d, f k r.. (a) 7 7 ; a, b a b (e) (f) (g) ; a, b a b ; a, b a b 7 7; a 7, b a b ; a, b a b
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