123 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

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1 3 PART : Solutions to Odd-Numbered Eercises and Practice Tests Section.7 Graphs of Rational Functions You should be able to graphf() - q()" (a) Find the - and -intercepts. (b) Find an vertical or horizontal asmptotes. (c) Plot additional points. (d) If the degree of the numerator is one more than the degree of the denominator, use long division to find the slant asmptote. Solutions to Odd-Numbered Eercises. g()=-+ 3. g()= -- Vertical shift one unit upward 5. g() = ~- Reflection in the -ais 7. g()- (- 7 Vertical shift two units downward 9o g() = ( + ) 3 Horizontal shift two units to the right. g()= 3 Horizontal Shift two units to the left - Reflection in the -ais

2 PART I: Solutions to Odd-Numbered Eercises and Practice Tests 3o f()- + -intercept: (0, -) Vertical asmptote: = Y C()- l + + l -intercept: 0) -intercept: (0, 5) Vertical asmptote: = - Horizontal asmptote: = C()! - 5 _7 3! 7, +5 g() Intercepts: (-~,0), (0,~) Vertical asmptote: = - Horizontal asmptote: = _! 5 Note: This is the graph off() - + (Eercise 3) shifted upward two units. 3 :z f() = Z -intercepts:(--~--,"f 0), (, 0 ) Vertical asmptote: = 0 Horizontal asmptote: = -ais smmetr - - _!

3 5 PART I: Solutions to Odd.Numbered Eercises and Practice Tests. f(x)-:z_ Intercept: (0, 0) Vertical asmptotes: =, = - Horizontal asmptote: = -ais smmetr..,.~. ~._~, -; -~ -if Y + +3 _+ 0 9! 0 3. f() - - Intercepts: (0, 0) Vertical asmptotes: =, = - Origin smmetr i Y 0!...~ - "~ Xo, o) 5. (+ ) g()--- (- ) Intercept: (-, 0) Vertical asmptotes: = 0 and =,, i Y 0-3 8, (-, , -8, I f(x) = a = ( + l)( - ) Intercept: (0, 0) Vertical asmptotes: = -, i i i I i,0) I ~ Y 0

4 PART I: Solutions to Odd-Numbered Eercises and Practice Tests 9. f() No intercepts [Note: f(0) is not defined] Vertical asmptote: = Horizontal asmptote: 5 No smmetr Y Y undef. undef f() Vertical asmptote: = Horizontal asmptote: = - - Domain: # or (-~~, ) U (, ~~) 3t+ 33. f(t) - t Vertical asmptote: t = 0 Horizontal asmptote: = h(t) - t + Domain: all real numbers OR (-~, ~~) I. - Domain: t = 0 or (-o~, O) U (0, o~) = +l +l 37. f() - - = (- 3)( + ) Domain: all real numbers ecept = 3, - Vertical asmptotes: = 3, = - 9

5 7 PART I: Solutions to Odd-Numbered Eercises and Practice Tests 0 i f() + ( ~ + ) Domain: all real numbers ecept 0, OR (-~~, O) U (0, Vertical asmptote: = 0. h()-- i i I i (I I,! lo There are two horizontal asmptotes: = g() = I- I.+ (- ) 5. f() --X lo There are two horizontal asmptotes: = +_. One vertical asmptote: = -. The graph crosses its horizontal asmptote: =. X+ 7. f()----~+ Vertical asmptote: = 0 Slant asmptote: = Origin smmetr ~ 9. h() ~ - - Intercept: (0, 0) Vertical asmptote: Slant asmptote: = +, 8 -, (o~ o),: / 8 sl. g() ~ + Intercept: (0, 0) Vertical asmptotes: = +_ Slant asmptotg: = ~ Origin smmetr

6 8 PART I: Solutions to Odd-Numbered Eercises and Practice Tests S3. f() = :z + = ~ + + ~ ~ + Intercepts: (-.59, 0), (0, ) Slant asmptote: = ~ (a) -intercept: (-, 0) +l (b) 0 - X--3 0=+l -l= (0, ) ~ -t ~ (a) -intercepts: (_+, O) (b) O=--. g() = X ~=l =+l = Domain:all real numbers ecept 0 OR o) u (o, Vertical asmptote: = 0 Slant asmptote: = =~ - + =-l+~ +l +l Domain: all real numbers ecept = - Vertical asmptote: ~ - Slant asmptote: = - \ 3. =~ (a) - (b) ( + 5) = -intercept: (-, 0) = -5 = 0 X "" --

7 9 PART I: Solutions to Odd-Numbered Eercises and Practice Tests 5, =-~ (a) - -intercept: (-, 0), (3~ 0) " 7.,(a) 0.5(0) () = C(0 + ) (c) C= 0+ C= 0+3 (0 + ) (+ i0) Domain: > 0 and < I000-0 Thus, 0 < < 990 OR [0, 990]. (b) 0 = - ~ X-- --X X-- = ( - ) O=-- 0 = ( + )(- 3) = -, = 3 9. (a) A = and (- )( - ) = =~ = Thus, A== - (X + - "" ~) (X - +) (b) Domain: Since the margins on the left and right are each inch, >, OR (, 95O As the tank is filled, the rate that the concentration is increasing slows down. It approaches the horizontal asmptote of C = ~ = The concentration reaches 7.5% when the tank is full ( = 990) ) 7. C= \ + l< O The area is minimum when. "~ 5.87 in. and ~.75 in. 3t z + t 73. C--- O<t :+50 (a) The horizontal asmptote is the t-ais, or C = 0. This indicates that the chemical eventuall dissipates. (b) The minimum occurs when ~ 0. ~ 0. o o The maimum occurs when t ~.5. (c) C < 0.35 when 0 < t <.5 hours and when t > 8.3 hours

8 30 PART I: Solutions to Odd-Numbered Eercises and Practice Tests 75. (a) 8,~0 (b) K = 38.9t O ~ 0 8,000 (c) K =.87F +.5t (d) Quadratic model is best because it fits the data well and is eas to use. Answer will var. 77. False, ou will have to lift our pencil to cross the vertical asmptote. - (3 - ) 79. h() Since h() is not reduced and (3 - ) is a factor of both the numerator and the denominator, = 3 is not a horizontal asmptote. There is a hole in the graph at = No, there are rational functions without vertical asmptotes. Two eamples are f()= 3 - (see Eercise 79) g(x) = Z + a 83. = - + ~ has slant asmptote = - and vertical asmptote at = -. We determine a so that has a zero at = 3: 0= ff==~a -7 a -7 ~ +- 5 Hence, = - + ~ = + + a -( + ) 85 = ( - 3) has vertical asmptote = 3 horizontal asmptote = - and zero at = =0:=~= ~ (-+)=-5-+~line

9 3 PART : Solutions to Odd-Numbered Eercises and Practice Tests 89. r- to=o = 0 =~ hodzontal line =0 - = =-~ line , lo -5 Semicircle Domain: - << Range: 0 < < Parabola Domain: all Range: < 9 Review Eercises for Chapter Solutions to Odd-Numbered Eercises. (a) = Vertical stretch (b) =- Vertical stretch and a reflection in the -ais _9 ~8 (e) = + Vertical shift two units upward 8 (d) =(+) Horizontal shift two units to the left

(4, 2) (2, 0) Vertical shift two units downward (2, 4) (0, 2) (1, 2) ( 1, 0) Horizontal shrink each x-value is multiplied by 1 2

(4, 2) (2, 0) Vertical shift two units downward (2, 4) (0, 2) (1, 2) ( 1, 0) Horizontal shrink each x-value is multiplied by 1 2 Section. Rational Functions 9. i i i i 9i. i i i. i 7i i i i i. 9 i9 i i. g f. g f. g f, ), ), ), ), ), ), ), ), ), ), ), ) Horizontal shift two units to the right Vertical shift two units downward Vertical