Section. Linear and Quadratic Functions and Modeling 6 Chapter Polnomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling Eploration. $000 per ear.. The equation will have the form v(t)=mt+b.the value of the building after 0 ear is v(0)=m(0)+b=b=0,000. The slope m is the rate of change, which is 000 (dollars per ear). So an equation for the value of the building (in dollars) as a function of the time (in ears) is v(t)= 000t+0,000.. v(0)=$0,000 and v(6)= 000(6)+0,000=8,000 dollars.. The equation v(t)=,000 becomes 000t+0,000=,000 000t=,000 t=. r Quick Review.. =8+.6. =.8-. -= - (+),or= 0.6+.8 (, ) 8 8. -= (-),or= + 6 (, ) (, ) 6. (-) =(-)(-)= --+6 = -8+6. (-6) =(-6)(-6)=(-8)(-6) = -8-8+8= -6+8 8. (+) = (+)(+) =( -)(+)= --- = --. -+=( -+)=(-)(-) =(-). ++=( ++)=(+)(+) =(+) Section. Eercises. Not a polnomial function because of the eponent. Polnomial of degree with leading coefficient. Polnomial of degree with leading coefficient. Polnomial of degree 0 with leading coefficient. Not a polnomial function because of the radical 6. Polnomial of degree with leading coefficient 8. m= so -= (-) f()= + (, ) 8 8. m= - so -= - (+) f()= - + (, ) (6, ) (, ) (, ). (+) =(+)(+)= +++ = +6+ Copright 0 Pearson Education, Inc.
68 Chapter Polnomial, Power, and Rational Functions. m= - so -6= - (+) f()= - + (, 6) (, ). m= so -= (-) f()= + (, ) 6. (f) The verte is in Quadrant I, at (, ), meaning it must be either (b) or (f). Since f(0)=, it cannot be (b): if the verte in (b) is (, ), then the intersection with the -ais occurs considerabl lower than (0, ). It must be (f).. (e) The verte is at (, ) in Quadrant IV, so it must be (e). 8. (c) The verte is at (, ) in Quadrant II and the parabola opens down, so it must be (c).. Translate the graph of f()= units right to obtain the graph of h()=(-), and translate this graph units down to obtain the graph of g()=(-) -. (, ). m= so -= (-0) f()= + (0, ) 0. Verticall shrink the graph of f()= b a factor of to obtain the graph of g()=, and translate this graph unit down to obtain the graph of h()= -. (, 0). m= so -= (-0) f()= + (0, ) (, 0). Translate the graph of f()= units left to obtain the graph of h()=(+), verticall shrink this graph b a factor of to obtain the graph of k()= (+), and translate this graph units down to obtain the graph of g()= (+) -.. (a) The verte is at (, ), in Quadrant III, eliminating all but (a) and (d). Since f(0)=, it must be (a).. (d) The verte is at (, ), in Quadrant III, eliminating all but (a) and (d). Since f(0)=, it must be (d).. (b) The verte is in Quadrant I, at (, ), meaning it must be either (b) or (f). Since f(0)=, it cannot be (f): if the verte in (f) is (, ), then the intersection with the -ais would be about (0, ). It must be (b). Copright 0 Pearson Education, Inc.
Section. Linear and Quadratic Functions and Modeling 6. Verticall stretch the graph of f()= b a factor of to obtain the graph of g()=, reflect this graph across the -ais to obtain the graph of k()=, and translate this graph up units to obtain the graph of h()= +.. h()= a + - b = a + # -+ + 6 b 8 = a + + b 8 Verte: a - ais: = -, 8 b;. f()=( -+)+6-=(-) +. Verte: (, ); ais: =; opens upward; does not intersect -ais. For #, with an equation of the form f()=a(-h) +k, the verte is (h, k) and the ais is =h.. Verte: (, ); ais: =. Verte: (, ); ais: =. Verte: (, ); ais: = 6. Verte: (, ); ais: =. f()=a + - b = -- =a + a + # - 6 + 6 b 6 b Verte: a - ; ais: = 6, - b 6 8. f()= a - - b = a - # -+ + 6 b 8 = ( - + ) 8 Verte: a ; ais: =, 8 b. f()= ( -8)+ = ( - # +6)+ +6= (-) + Verte: (, ); ais: = 0. f()= a - +6 b = +6- = a - a - # + + 6 b b [, 6] b [0, 0]. g()=( -6+)+-=(-) +. Verte: (, ); ais: =; opens upward; does not intersect -ais. [, 8] b [0, 0]. f()= ( +6)+ = ( +6+6)++6= (+8) +. Verte: ( 8, ); ais: = 8; opens downward; intersects -ais at about 6.60 and 0.60(-8 ; ). [ 0, ] b [ 0, 0] 6. h()= ( -)+8= ( -+)+8+ = (-) + Verte: (, ); ais: =; opens downward; intersects -ais at and. Verte: a ais: =, b;. g()= a - 6 + b = +- = a - a - # + + b b Verte: a ais: =, b; [, ] b [ 0, ] Copright 0 Pearson Education, Inc.
0 Chapter Polnomial, Power, and Rational Functions. f()=( +)+ = +- = a + a + + +. b b Verte: a - ais: = - opens upward; does not, b; ;. Weak positive 8. No correlation. (a) intersect the -ais; verticall stretched b. [, ] b [0, 0] (b) Strong, positive, linear; r=0.8 0. (a) [., ] b [,.] 8. g()=( -)+ = a - + +- b =a - -. b Verte: a ais: = opens upward; intersects, - b; ; -ais at about 0.8 and.6 aor verticall stretched b. ; 8b; [, ] b [ 0, 0] For #, use the form =a(-h) +k, taking the verte (h, k) from the graph or other given information.. h= and k=, so =a(+) -. Now substitute =, = to obtain =a-, so a=: =(+) -. 0. h= and k=, so =a(-) -. Now substitute =0, = to obtain =a-, so a=: =(-) -.. h= and k=, so =a(-) +. Now substitute =, = to obtain =a+, so a= : = (-) +.. h= and k=, so =a(+) +. Now substitute =, = to obtain =a+, so a= : = (+) +.. h= and k=, so =a(-) +. Now substitute =0, = to obtain =a+, so a=: =(-) +.. h= and k=, so =a(+) -. Now substitute =, = to obtain =a-, so a= - = - (+) -. :. Strong positive 6. Strong negative [0, 80] b [0, 80] (b) Strong, negative, linear; r= 0.. m= - 0 = 0 and b=0, so v(t)= 0t+0. At t=, v()=( 0)()+0=$0.. Let be the number of dolls produced each week and be the average weekl costs. Then m=.0, and b=0, so =.0+0, or 00=.0+0: =; dolls are produced each week.. (a) L 0.8 +.6. The slope, m L 0.8, represents the average annual increase in fuel econom for light dut trucks, about 0. mpg per ear. (b) Setting = in the regression equation leads to L mpg.. If the length is, then the width is 0-, so A()=(0-); maimum of 6 ft when = (the dimensions are ft* ft).. (a) [0, 0] b [0, 00] is one possibilit. (b) When. or.66 either, units or,66 units. 6. The area of the picture and the frame, if the width of the picture is ft, is A()=(+)(+) ft. This equals 08 when =, so the painting is ft* ft.. If the strip is feet wide, the area of the strip is A()=(+)(0+)-00 ft. This equals 0 ft when =. ft. 8. (a) R()=(800+0)(00-). (b) [0, ] b [00,000, 60,000] is one possibilit (shown). [0, ] b [00,000, 60,000] Copright 0 Pearson Education, Inc.
Section. Linear and Quadratic Functions and Modeling (c) The maimum income $0,000 is achieved when =, corresponding to rent of $0 per month.. (a) R()=(6,000-00)(0.0+0.0). (b) Man choices of Xma and Ymin are reasonable. Shown is [0, ] b [,000,,000]. 6. (a) h= 6t +80t-. The graph is shown in the window [0, ] b [, 0]. [0, ] b [000, 000] (c) The maimum revenue $6,00 is achieved when =8; that is, charging 0 cents per can. 60. Total sales would be S()=(0+)(0-) thousand dollars, when additional salespeople are hired. The maimum occurs when = (halfwa between the two zeros, at 0 and 0). 6. (a) g L ft/sec. s 0 = 8 ft and v 0 = ft/sec. So the models are height= s(t) = -6t + t + 8 and vertical velocit= v(t) = -t +. The maimum height occurs at the verte of s(t). h = - b and a = - (-6) =.8, k = s(.8) =.. The maimum height of the baseball is about ft above the field. (b) The amount of time the ball is in the air is a zero of s(t). Using the quadratic formula, we obtain t = - ; - (-6)(8) (-6) - ;,6 = L -0. or 6.. Time is not - negative, so the ball is in the air about 6. seconds. (c) To determine the ball s vertical velocit when it hits the ground, use v(t) = -t +, and solve for t = 6.. v(6.) = -(6.) + L - ft/sec when it hits the ground. 6. (a) h= 6t +8t+.. (b) The maimum height is. ft,. sec after it is thrown. [0, ] b [, 0] (b) The maimum height is 0 ft,. sec after it is shot. 6. The eact answer is, or about.6 ft/sec. In addition to the guess-and-check strateg suggested, this can be found algebraicall b noting that the verte of the parabola =a +b+c has coordinate b c- (note a= 6 and c=0), and setting a = b 6 this equal to 8. 6. The quadratic regression is 0. +.6+.6. Plot this curve together with the curve =0, and then find the intersection to find when the number of patent applications will reach 0,000. Note that we use =0 because the data were given as a number of thousands. The intersection occurs at L.0, so the number of applications will reach 0,000 approimatel ears after 80 in 00. 6 ft 66. (a) m= =0.06 0 ft (b) r 6 ft, or about 0. mi. (c).6 ft 6. (a) [, ] b [0, 0] (b) 0.68+.0 (c) On average, the children gained 0.68 lb per month. (d) [, ] b [0, 0] [0,.] b [0, ] (e). lbs 68. (a) The linear regression is L. + 0., where represents the number of ears since 0. (b) 00 is 80 ears after 0, so substitute 80 into the equation to predict the median income of women in 00. =.(80) + 0. L $,. Copright 0 Pearson Education, Inc.
Chapter Polnomial, Power, and Rational Functions 6. The Identit Function f()= [.,.] b [.,.] Domain: (- q, q) Range: (- q, q) Continuit: The function is continuous on its domain. Increasing decreasing behavior: Increasing for all Smmetr: Smmetric about the origin Boundedness: Not bounded Local etrema: None Horizontal asmptotes: None Vertical asmptotes: None End behavior: lim f() = -q and lim f() = q : -q : q 0. The Squaring Function f()= [.,.] b [, ] Domain: (- q, q) Range: [0, q) Continuit: The function is continuous on its domain. Increasing decreasing behavior: Increasing on [0, q), decreasing on (- q, 0]. Smmetr: Smmetric about the -ais Boundedness: Bounded below, but not above Local etrema: Local minimum of 0 at =0 Horizontal asmptotes: None Vertical asmptotes: None End behavior: lim f() = lim f() = q : -q : q. False. For f()= +-, the initial value is f(0)=.. True. B completing the square, we can rewrite f() so that f()= a - + +- b = a - + Since f() Ú f()>0 for all. b., -. m = The answer is E. - (-) = - 6 = -.. f()=m+b = - ( )+b = +b b=- =. The answer is C. For # 6, f()=(+) - corresponds to f()=a(-h) +k with a= and (h, k)=(, ).. The ais of smmetr runs verticall through the verte: =. The answer is B. 6. The verte is (h, k)=(, ). The answer is E.. (a) Graphs (i), (iii), and (v) are linear functions. The can all be represented b an equation =a+b, where a Z 0. (b) In addition to graphs (i), (iii), and (v), graphs (iv) and (vi) are also functions, the difference is that (iv) and (vi) are constant functions, represented b =b, b Z 0. (c) (ii) is not a function because a single value (i.e., = ) results in a multiple number of -values. In fact, there are infinitel man -values that are valid for the equation =. f() - f() - 8. (a) = = - f() - f() - (b) = = - f(c) - f(a) c - a ()(c + a) (c) = = =c+a g() - g() - (d) = = - g() - g() - (e) = = - g(c) - g(a) (c + ) - (a + ) (f) = c - a = = h(c) - h(a) (c - ) - (a - ) (g) = c - a = = k(c) - k(a) (mc + b) - (ma + b) (h) = mc - ma = =m l(c) - l(a) c - a (i) = = -b a = -b a = -b a ()(c + ac + a ) = =c +ac+a (). The line that minimizes the sum of the squares of vertical distances is nearl alwas different from the line that minimizes the sum of the squares of horizontal distances to the points in a scatter plot. For the data in Table., the regression line obtained from reversing the ordered pairs has a slope of - ; whereas, the inverse of the,.0 function in Eample has a slope of - close,8. but not the same slope. Copright 0 Pearson Education, Inc.
Section. Power Functions with Modeling 80. (a) [0, ] b [, 6] (b) 0.+8. [0, ] b [, 6] (c) 0.6+6.0 [0, ] b [0, ] (d) The median median line appears to be the better fit, because it approimates more of the data values more closel. 8. (a) If a +b +c=0, then -b ; b - ac = b the quadratic a -b + b - ac formula. Thus, = and a -b - b - ac = and a -b + b - ac - b - b - ac + = a -b = a = -b a = -b a. (b) Similarl, # = a -b + b - ac ba -b - b - ac b a a b - (b - ac) ac c = = = a a a. 8. f()=(-a)(-b)= -b-a+ab = +( a-b)+ab. If we use the verte form of a quadratic function, we have h= - a -a - b b a + b =. The ais is =h= a + b. 8. Multipl out f() to get -(a+b)+ab. Complete (a + b) the square to get a - a + b b +ab-. The a + b verte is then (h, k) where h= and (a + b) (a - b) k=ab- = -. 8. and are given b the quadratic formula -b ; b - ac a ; then + = - b and the line of a, smmetr is = - b which is eactl equal to a, 8. The Constant Rate of Change Theorem states that a function defined on all real numbers is a linear function if and onl if it has a constant nonzero average rate of change between an two points on its graph. To prove this, suppose f() = m + b with m and b constants and m Z 0. Let and be real numbers with Z. Then the average rate of change is f( ) - f( ) = (m + b) - (m + b) = - - m - m = m( - ) = m, a nonzero constant. - - Now suppose that m and are constants, with m Z 0. Let be a real number such that Z, and let f be a function defined on all real numbers such that f() - f( ) = m. Then f() - f( ) = m( - ) = - m - m, and f() = m + (f( ) - m ). f( ) - m is a constant; call it b. Then f( ) - m = b; so, f( ) = b + m and f() = b + m for all Z. Thus, f is a linear function. Section. Power Functions with Modeling Eploration. [.,.] b [.,.] [, ] b [, ] +. Copright 0 Pearson Education, Inc.