1. m = 3, P (3, 1) 2. m = 2, P ( 5, 8) 3. m = 1, P ( 7, 1) 4. m = m = 0, P (3, 117) 8. m = 2, P (0, 3)

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1 . Linear Functions 69.. Eercises To see all of the help resources associated with this section, click OSttS Chapter. In Eercises - 0, find both the point-slope form and the slope-intercept form of the line with the given slope which passes through the given point. For help with these eercises, click on one of the resources below: Finding the equation of a line Writing a line in slope-intercept form. m =, P (, ). m =, P ( 5, 8). m =, P ( 7, ) 4. m =, P (, ) 5. m = 5, P (0, 4) 6. m = 7, P (, 4) 7. m = 0, P (, 7) 8. m =, P (0, ) 9. m = 5, P (, ) 0. m = 678, P (, ) In Eercises - 0, find the slope-intercept form of the line which passes through the given points. For help with these eercises, click on one of the resources below: Finding slope Finding the equation of a line Writing a line in slope-intercept form. P (0, 0), Q(, 5). P (, ), Q(, ). P (5, 0), Q(0, 8) 4. P (, 5), Q(7, 4) 5. P (, 5), Q(7, 5) 6. P (4, 8), Q(5, 8) 7. P (, ( 4), Q 5, 7 ) 4 8. P (, 7 ) (, Q, ) 9. P (, ), Q (, ) 0. P (, ), Q (, )

2 70 Linear and Quadratic Functions In Eercises - 6, graph the function. Find the slope, -intercept and -intercept, if an eist. For help with these eercises, click on one of the resources below: Writing a line in slope-intercept form Graphing a line in slope-intercept form. f() =. f() =. f() = 4. f() = 0 5. f() = + 6. f() = 7. Find all of the points on the line = + which are 4 units from the point (, ). For help with Eercises 8-6, click on the resourse below: Writing linear models Interpreting slope as a rate of change 8. Jeff can walk comfortabl at miles per hour. Find a linear function d that represents the total distance Jeff can walk in t hours, assuming he doesn t take an breaks. 9. Carl can stuff 6 envelopes per minute. Find a linear function E that represents the total number of envelopes Carl can stuff after t hours, assuming he doesn t take an breaks. 0. A landscaping compan charges $45 per cubic ard of mulch plus a deliver charge of $0. Find a linear function which computes the total cost C (in dollars) to deliver cubic ards of mulch.. A plumber charges $50 for a service call plus $80 per hour. If she spends no longer than 8 hours a da at an one site, find a linear function that represents her total dail charges C (in dollars) as a function of time t (in hours) spent at an one given location.. A salesperson is paid $00 per week plus 5% commission on her weekl sales of dollars. Find a linear function that represents her total weekl pa, W (in dollars) in terms of. What must her weekl sales be in order for her to earn $ for the week?. An on-demand publisher charges $.50 to print a 600 page book and $5.50 to print a 400 page book. Find a linear function which models the cost of a book C as a function of the number of pages p. Interpret the slope of the linear function and find and interpret C(0). 4. The Topolog Tai Compan charges $.50 for the first fifth of a mile and $0.45 for each additional fifth of a mile. Find a linear function which models the tai fare F as a function of the number of miles driven, m. Interpret the slope of the linear function and find and interpret F (0).

3 . Linear Functions 7 5. Water freezes at 0 Celsius and Fahrenheit and it boils at 00 C and F. (a) Find a linear function F that epresses temperature in the Fahrenheit scale in terms of degrees Celsius. Use this function to convert 0 C into Fahrenheit. (b) Find a linear function C that epresses temperature in the Celsius scale in terms of degrees Fahrenheit. Use this function to convert 0 F into Celsius. (c) Is there a temperature n such that F (n) = C(n)? 6. Legend has it that a bull Sasquatch in rut will howl approimatel 9 times per hour when it is 40 F outside and onl 5 times per hour if it s 70 F. Assuming that the number of howls per hour, N, can be represented b a linear function of temperature Fahrenheit, find the number of howls per hour he ll make when it s onl 0 F outside. What is the applied domain of this function? Wh? 7. Economic forces beond anone s control have changed the cost function for PortaBos to C() = Rework Eample..5 with this new cost function. 8. In response to the economic forces in Eercise 7 above, the local retailer sets the selling price of a PortaBo at $50. Remarkabl, 0 units were sold each week. When the sstems went on sale for $0, 40 units per week were sold. Rework Eamples..6 and..7 with this new data. What difficulties do ou encounter? 9. A local pizza store offers medium two-topping pizzas delivered for $6.00 per pizza plus a $.50 deliver charge per order. On weekends, the store runs a game da special: if si or more medium two-topping pizzas are ordered, the are $5.50 each with no deliver charge. Write a piecewise-defined linear function which calculates the cost C (in dollars) of p medium two-topping pizzas delivered during a weekend. 40. A restaurant offers a buffet which costs $5 per person. For parties of 0 or more people, a group discount applies, and the cost is $.50 per person. Write a piecewise-defined linear function which calculates the total bill T of a part of n people who all choose the buffet. 4. A mobile plan charges a base monthl rate of $0 for the first 500 minutes of air time plus a charge of 5 for each additional minute. Write a piecewise-defined linear function which calculates the monthl cost C (in dollars) for using m minutes of air time. HINT: You ma want to revisit Eercise 74 in Section.4 4. The local pet shop charges per cricket up to 00 crickets, and 0 per cricket thereafter. Write a piecewise-defined linear function which calculates the price P, in dollars, of purchasing c crickets. 4. The cross-section of a swimming pool is below. Write a piecewise-defined linear function which describes the depth of the pool, D (in feet) as a function of: (a) the distance (in feet) from the edge of the shallow end of the pool, d.

4 7 Linear and Quadratic Functions (b) the distance (in feet) from the edge of the deep end of the pool, s. (c) Graph each of the functions in (a) and (b). Discuss with our classmates how to transform one into the other and how the relate to the diagram of the pool. d ft. 7 ft. s ft. ft. 8 ft. 0 ft. 5 ft. In Eercises 44-49, compute the average rate of change of the function over the specified interval. For help with these eercises, click on the resource below: Finding the average rate of change of a function 44. f() =, [, ] 45. f() =, [, 5] 46. f() =, [0, 6] 47. f() =, [, ] 48. f() = + 4, [5, 7] 49. f() = + 7, [ 4, ] In Eercises 50-5, compute the average rate of change of the given function over the interval [, + h]. Here we assume [, + h] is in the domain of the function. 50. f() = 5. f() = 5. f() = f() = The height of an object dropped from the roof of an eight stor building is modeled b: h(t) = 6t + 64, 0 t. Here, h is the height of the object off the ground in feet, t seconds after the object is dropped. Find and interpret the average rate of change of h over the interval [0, ]. 55. Using data from Bureau of Transportation Statistics, the average fuel econom F in miles per gallon for passenger cars in the US can be modeled b F (t) = t t + 6, 0 t 8, where t is the number of ears since 980. Find and interpret the average rate of change of F over the interval [0, 8].

5 . Linear Functions The temperature T in degrees Fahrenheit t hours after 6 AM is given b: T (t) = t + 8t +, 0 t (a) Find and interpret T (4), T (8) and T (). (b) Find and interpret the average rate of change of T over the interval [4, 8]. (c) Find and interpret the average rate of change of T from t = 8 to t =. (d) Find and interpret the average rate of temperature change between 0 AM and 6 PM. 57. Suppose C() = represents the costs, in hundreds, to produce thousand pens. Find and interpret the average rate of change as production is increased from making 000 to 5000 pens. 58. With the help of our classmates find several other real-world eamples of rates of change that are used to describe non-linear phenomena. (Parallel Lines) Recall from Intermediate Algebra that parallel lines have the same slope. (Please note that two vertical lines are also parallel to one another even though the have an undefined slope.) In Eercises 59-64, ou are given a line and a point which is not on that line. Find the line parallel to the given line which passes through the given point. 59. = +, P (0, 0) 60. = 6 + 5, P (, ) 6. = 7, P (6, 0) 6. = 4, P (, ) 6. = 6, P (, ) 64. =, P ( 5, 0) (Perpendicular Lines) Recall from Intermediate Algebra that two non-vertical lines are perpendicular if and onl if the have negative reciprocal slopes. That is to sa, if one line has slope m and the other has slope m then m m =. (You will be guided through a proof of this result in Eercise 7.) Please note that a horizontal line is perpendicular to a vertical line and vice versa, so we assume m 0 and m 0. In Eercises 65-70, ou are given a line and a point which is not on that line. Find the line perpendicular to the given line which passes through the given point. 65. = +, P (0, 0) 66. = 6 + 5, P (, ) 67. = 7, P (6, 0) 68. = 4, P (, ) 69. = 6, P (, ) 70. =, P ( 5, 0)

6 74 Linear and Quadratic Functions 7. We shall now prove that = m + b is perpendicular to = m + b if and onl if m m =. To make our lives easier we shall assume that m > 0 and m < 0. We can also move the lines so that their point of intersection is the origin without messing things up, so we ll assume b = b = 0. (Take a moment with our classmates to discuss wh this is oka.) Graphing the lines and plotting the points O(0, 0), P (, m ) and Q(, m ) gives us the following set up. P O Q The line = m will be perpendicular to the line = m if and onl if OP Q is a right triangle. Let d be the distance from O to P, let d be the distance from O to Q and let d be the distance from P to Q. Use the Pthagorean Theorem to show that OP Q is a right triangle if and onl if m m = b showing d + d = d if and onl if m m =. 7. Show that if a b, the line containing the points (a, b) and (b, a) is perpendicular to the line =. (Coupled with the result from Eample..7 on page, we have now shown that the line = is a perpendicular bisector of the line segment connecting (a, b) and (b, a). This means the points (a, b) and (b, a) are smmetric about the line =. We will revisit this smmetr in section 5..) 7. The function defined b I() = is called the Identit Function. (a) Discuss with our classmates wh this name makes sense. (b) Show that the point-slope form of a line (Equation.) can be obtained from I using a sequence of the transformations defined in Section.7.

7 . Linear Functions 75 Checkpoint Quiz.. The US Unemploment Rate was 5.0% in Januar 008 and 7.7% in Januar (a) Find a linear function which fits these data using the number of ears since Januar 008, t, as the independent variable and the unemploment rate, U, as the dependent variable. (b) Find a reasonable applied domain for the model ou found in (a). (c) Use this model to predict the unemploment rate in Januar 00. NOTE: The actual unemploment rate was 9.7%.. The height of a frog off the ground, h (in feet) t seconds after it jumps in the air is given b h(t) = 6t + t, 0 t. Find and interpret the average rate of change of h over the interval [, ]. For worked out solutions to this quiz, click the links below: Quiz Solution Part Quiz Solution Part 8 Source:

8 76 Linear and Quadratic Functions.. Answers. + = ( ) = 0. + = ( + 7) = 8. 8 = ( + 5) = 4. = ( + ) = = 5 ( 0) 6. 4 = = ( + ) = = 0 = 7 9. = 5( ) = = ( 0) = 0. + = 678( + ) = = 5. =. = = = 5 6. = 8 7. = = = 0. =. f() = slope: m = -intercept: (0, ) -intercept: (, 0). f() = slope: m = -intercept: (0, ) -intercept: (, 0) 4 4

9 . Linear Functions 77. f() = slope: m = 0 -intercept: (0, ) -intercept: none 4 4. f() = 0 slope: m = 0 -intercept: (0, 0) -intercept: {(, 0) is a real number} 5. f() = + slope: m = -intercept: ( 0, ) -intercept: (, 0) 6. f() = slope: m = -intercept: ( 0, ) -intercept: (, 0) 7. (, ) and ( 5, 7 ) 5 8. d(t) = t, t E(t) = 60t, t C() = , 0.. C(t) = 80t + 50, 0 t 8.. W () = , 0 She must make $5500 in weekl sales.. C(p) = 0.05p +.5 The slope 0.05 means it costs.5 per page. C(0) =.5 means there is a fied, or start-up, cost of $.50 to make each book. 4. F (m) =.5m +.05 The slope.5 means it costs an additional $.5 for each mile beond the first 0. miles. F (0) =.05, so according to the model, it would cost $.05 for a trip of 0 miles. Would this ever reall happen? Depends on the driver and the passenger, we suppose.

10 78 Linear and Quadratic Functions 5. (a) F (C) = 9 5 C + (b) C(F ) = 5 9 (F ) = 5 9 F 60 9 (c) F ( 40) = 40 = C( 40). 6. N(T ) = 5 T + 4 Having a negative number of howls makes no sense and since N(07.5) = 0 we can put an upper bound of 07.5 F on the domain. The lower bound is trickier because there s nothing other than common sense to go on. As it gets colder, he howls more often. At some point it will either be so cold that he freezes to death or he s howling non-stop. So we re going to sa that he can withstand temperatures no lower than 60 F so that the applied domain is [ 60, 07.5]. { 6p +.5 if p 5 9. C(p) = 5.5p if p 6 { 5n if n T (n) =.5n if n 0 { 0 if 0 m C(m) = (m 500) if m > 500 { 0.c if c P (c) = + 0.(c 00) if c > (a) (b) (c) 8 if 0 d 5 D(d) = d + if 5 d 7 if 7 d 7 D(s) = if 0 s 0 s if 0 s 8 if s = D(d) 0 7 = D(s)

11 . Linear Functions ( ) = ( ) = = h + h ( )( + h ) 5 5 = 5 ( ) ( ) = 0 (() + () 7) (( 4) + ( 4) 7) ( 4) ( + h) h The average rate of change is h() h(0) 0 =. During the first two seconds after it is dropped, the object has fallen at an average rate of feet per second. (This is called the average velocit of the object.) F (8) F (0) 55. The average rate of change is 8 0 = 0.7. During the ears from 980 to 008, the average fuel econom of passenger cars in the US increased, on average, at a rate of 0.7 miles per gallon per ear. 56. (a) T (4) = 56, so at 0 AM (4 hours after 6 AM), it is 56 F. T (8) = 64, so at PM (8 hours after 6 AM), it is 64 F. T () = 56, so at 6 PM ( hours after 6 AM), it is 56 F. T (8) T (4) (b) The average rate of change is 8 4 =. Between 0 AM and PM, the temperature increases, on average, at a rate of F per hour. T () T (8) (c) The average rate of change is 8 =. Between PM and 6 PM, the temperature decreases, on average, at a rate of F per hour. T () T (4) 4 = 0. Between 0 AM and 6 PM, the tempera- (d) The average rate of change is ture, on average, remains constant. = The average rate of change is C(5) C() 5 =. As production is increased from 000 to 5000 pens, the cost decreases at an average rate of $00 per 000 pens produced (0 per pen.) 59. = 60. = = 4 6. = 6. = 64. = = 66. = = = = 70. = 0