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13.2.2: Chapter 2

  • Page ID
    117277
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    Try It

    2.1 Linear Functions

    1.

    m= 43 02 = 1 2 = 1 2 m= 43 02 = 1 2 = 1 2 ; decreasing because m<0. m<0.

    2.

    m= 1,8681,442 2,0122,009 = 426 3 =142 people per year m= 1,8681,442 2,0122,009 = 426 3 =142 people per year

    3.

    y2=2( x+2 ) y2=2( x+2 ) ; y=2x2 y=2x2

    4.

    y0=3( x0 ) y0=3( x0 ) ; y=3x y=3x

    5.

    y=7x+3 y=7x+3

    6.

    H( x )=0.5x+12.5 H( x )=0.5x+12.5

    2.2 Graphs of Linear Functions

    1.
    cfc232588c0a93dd535cca1d0a7d141ac06b41cd
    2.

    Possible answers include (3,7),(3,7), (6,9),(6,9), or (9,11).(9,11).

    3.
    a030ae4fc6ac6124a12d0a37510ada60f6c19c19
    4.

    (16, 0)(16, 0)

    5.
    1. f(x)=2xf(x)=2x
    2. g(x)= 1 2 x g(x)= 1 2 x
    6.

    y=13x+6y=13x+6

    7.
    1. (0,5)(0,5)
    2. (5, 0)(5, 0)
    3. Slope -1
    4. Neither parallel nor perpendicular
    5. Decreasing function
    6. Given the identity function, perform a vertical flip (over the t-axis) and shift up 5 units.

    2.3 Modeling with Linear Functions

    1.
    1. C( x )=0.25x+25,000 C( x )=0.25x+25,000
    2. The y-intercept is ( 0,25,000 ). ( 0,25,000 ). If the company does not produce a single doughnut, they still incur a cost of $25,000.
    2.
    1. 41,100
    2. 2020
    3.

    21.57 miles

    2.4 Fitting Linear Models to Data

    1.

    54°F54°F

    2.

    150.871 billion gallons; extrapolation

    2.1 Section Exercises

    1.

    Terry starts at an elevation of 3000 feet and descends 70 feet per second.

    3.

    3 miles per hour

    5.

    d( t )=10010t d( t )=10010t

    7.

    Yes.

    9.

    No.

    11.

    No.

    13.

    No.

    15.

    Increasing.

    17.

    Decreasing.

    19.

    Decreasing.

    21.

    Increasing.

    23.

    Decreasing.

    25.

    3

    27.

    1313

    29.

    4545

    31.

    f(x)=12x+72f(x)=12x+72

    33.

    y=2x+3y=2x+3

    35.

    y=13x+223y=13x+223

    37.

    y=45x+4y=45x+4

    39.

    5454

    41.

    y= 2 3 x+1 y= 2 3 x+1

    43.

    y=2x+3 y=2x+3

    45.

    y=3 y=3

    47.

    Linear, g(x)=3x+5g(x)=3x+5

    49.

    Linear, f(x)=5x5f(x)=5x5

    51.

    Linear, g(x)=252x+6g(x)=252x+6

    53.

    Linear, f(x)=10x24f(x)=10x24

    55.

    f(x)=58x+17.3 f(x)=58x+17.3

    57.
    037e6ae65a278bdd4e7975e5f766ddcf91d0b538
    59.

    a. a=11,900a=11,900; b=1000.1b=1000.1 b. q(p)=1000p100q(p)=1000p100

    61.
    ..
    63.

    x=163x=163

    65.

    x=ax=a

    67.

    y=dcaxadcay=dcaxadca

    69.

    $45 per training session.

    71.

    The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24.

    73.

    The slope is 400.400. This means for every year between 1960 and 1989, the population dropped by 400 per year in the city.

    75.

    c.

    2.2 Section Exercises

    1.

    The slopes are equal; y-intercepts are not equal.

    3.

    The point of intersection is (a,a).(a,a). This is because for the horizontal line, all of the yy coordinates are aa and for the vertical line, all of the xx coordinates are a.a. The point of intersection will have these two characteristics.

    5.

    First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y=mx+by=mx+b and solve for b.b. Then write the equation of the line in the form y=mx+by=mx+b by substituting in mm and b.b.

    7.

    neither parallel or perpendicular

    9.

    perpendicular

    11.

    parallel

    13.

    (2, 0)(2, 0); (0, 4)(0, 4)

    15.

    (15, 0)(15, 0); (0, 1)(0, 1)

    17.

    (8, 0)(8, 0); (0, 28)(0, 28)

    19.

    Line 1:m=8 Line 1:m=8
    Line 2:m=6Line 2:m=6
    Neither Neither

    21.

    Line 1:m= 1 2 Line 1:m= 1 2
    Line 2:m=2Line 2:m=2
    Perpendicular Perpendicular

    23.

    Line 1:m=2 Line 1:m=2
    Line 2:m=2Line 2:m=2
    Parallel Parallel

    25.

    g(x)=3x3g(x)=3x3

    27.

    p(t)=13t+2p(t)=13t+2

    29.

    (2,1)(2,1)

    31.

    (175,53)(175,53)

    33.

    F

    35.

    C

    37.

    A

    39.
    7f3c7454fb27900b42c8569b8d28a8d6aa8af8ec
    41.
    1da664f94359eede4623734794bed8a986d57ee3
    43.
    237172d9456a36429ac0a55d0aacfb25d0de3a86
    45.
    4a85be25e05eba47e32b306e935d8e65a1526559
    47.
    b6b5b31ba81b16770e866b57b427d3acafe5468a
    49.
    19a81cb20594ac6814c0511ef943f7ea6f6cdebb
    51.
    0e75777001d2f9e1559541346e75e8bfb63d36e2
    53.
    d4b7b46b426992b131010bf8ecd13dfd4b29135b
    55.
    30797a72091582d1626758acccd6f855b68dad4c
    57.
    2453cab026a83f7ebaf0444fdc76303a3c602472
    59.
    1. g(x)=0.75x5.5g(x)=0.75x5.5
    2. 0.75
    3. (0,5.5)(0,5.5)
    61.

    y=3y=3

    63.

    x=3x=3

    65.

    no point of intersection

    67.

    (2, 7) (2, 7)

    69.

    (10, –5)(10, –5)

    71.

    y=100x98 y=100x98

    73.

    x< 1999 201 x> 1999 201 x< 1999 201 x> 1999 201

    75.

    Less than 3000 texts

    2.3 Section Exercises

    1.

    Determine the independent variable. This is the variable upon which the output depends.

    3.

    To determine the initial value, find the output when the input is equal to zero.

    5.

    6 square units

    7.

    20.012 square units

    9.

    2,300

    11.

    64,170

    13.

    P( t )=75,000+2,500t P( t )=75,000+2,500t

    15.

    (–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

    17.

    Ten years after the model began.

    19.

    W( t )=0.5t+7.5 W( t )=0.5t+7.5

    21.

    ( 15,0 ) ( 15,0 ) : The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. ( 0, 7.5 ) ( 0, 7.5 ) : The baby weighed 7.5 pounds at birth.

    23.

    At age 5.8 months.

    25.

    C( t )=12,025205t C( t )=12,025205t

    27.

    (58.7, 0) (58.7, 0) : In roughly 59 years, the number of people inflicted with the common cold would be 0. (0,12,025) (0,12,025) : Initially there were 12,025 people afflicted by the common cold.

    29.

    2064

    31.

    y=2t+180 y=2t+180

    33.

    In 2070, the company’s profit will be zero.

    35.

    y=30t300 y=30t300

    37.

    (10, 0) In 1990, the profit earned zero profit.

    39.

    Hawaii

    41.

    During the year 1933

    43.

    $105,620

    45.
    1. 696 people
    2. 4 years
    3. 174 people per year
    4. 305 people
    5. P(t)=305+174t P(t)=305+174t
    6. 2,219 people
    47.
    1. C( x )=0.15x+10 C( x )=0.15x+10
    2. The flat monthly fee is $10 and there is an additional $0.15 fee for each additional minute used
    3. $113.05
    49.
    1. P( t )=190t+4360 P( t )=190t+4360
    2. 6,640 moose
    51.
    1. R( t )=162.1t R( t )=162.1t
    2. 5.5 billion cubic feet
    3. During the year 2017
    53.

    More than 133 minutes

    55.

    More than $42,857.14 worth of jewelry

    57.

    $66,666.67

    2.4 Section Exercises

    1.

    When our model no longer applies, after some value in the domain, the model itself doesn’t hold.

    3.

    We predict a value outside the domain and range of the data.

    5.

    The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.

    7.

    61.966 years

    9.

    No.

    11.

    No.

    13.

    Interpolation. About 60° F. 60° F.

    15.

    C

    17.

    B

    19.
    71e838bf549242004a7f1eecfeeb66cde279b324
    21.
    56b1e6dea853bbafc844b667aaf19b77f770ff79
    23.

    Yes, trend appears linear because r=0.985r=0.985 and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.

    25.

    y=1.640x+13.800y=1.640x+13.800, r=0.987r=0.987

    27.

    y=0.962x+26.86,r=0.965 y=0.962x+26.86,r=0.965

    29.

    y=1.981x+60.197y=1.981x+60.197; r=0.998r=0.998

    31.

    y=0.121x38.841,r=0.998y=0.121x38.841,r=0.998

    33.

    (−2,−6),(1,−12),(5,−20),(6,−22),(9,−28)(−2,−6),(1,−12),(5,−20),(6,−22),(9,−28); y=−2x−10y=−2x−10

    35.

    (189.8,0)(189.8,0) If 18,980 units are sold, the company will have a profit of zero dollars.

    37.

    y=0.00587x+1985.41y=0.00587x+1985.41

    39.

    y=20.25x671.5y=20.25x671.5

    41.

    y=10.75x+742.50y=10.75x+742.50

    Review Exercises

    1.

    Yes

    3.

    Increasing.

    5.

    y=3x+26y=3x+26

    7.

    3

    9.

    y=2x2y=2x2

    11.

    Not linear.

    13.

    parallel

    15.

    (–9,0);(0,–7)(–9,0);(0,–7)

    17.

    Line 1: m=2;m=2; Line 2: m=2;m=2; Parallel

    19.

    y=0.2x+21y=0.2x+21

    21.
    09f3d933b61b5a18631e559c4be70c86977de017
    23.

    250.

    25.

    118,000.

    27.

    y=300x+11,500y=300x+11,500

    29.

    a) 800; b) 100 students per year; c) P(t)=100t+1700P(t)=100t+1700

    31.

    18,500

    33.

    $91,625

    35.

    Extrapolation.

    8beebd5bb1c6ca6c8de43339354cf3293128697c
    37.
    7b9b0a87d04e3bb08bc86beddd2023daa447fb01
    39.

    Midway through 2024.

    41.

    y=1.294x+49.412;r=0.974y=1.294x+49.412;r=0.974

    43.

    Early in 2022

    45.

    7,660

    Practice Test

    1.

    Yes.

    3.

    Increasing

    5.

    y=−1.5x6y=−1.5x6

    7.

    y=2x1y=2x1

    9.

    No.

    11.

    Perpendicular

    13.

    (7,0)(7,0); (0,2)(0,2)

    15.

    y=0.25x+12y=0.25x+12

    17.
    966bcb933e3477ba02be3f6ef48eeea647d12a76
    19.

    150

    21.

    165,000

    23.

    y=875x+10,675y=875x+10,675

    25.

    a) 375; b) dropped an average of 46.875, or about 47 people per year; c) y=46.875t+1250y=46.875t+1250

    27.
    0bbdc170ab786b641a5f08bcbf62e42d62d57ebd
    29.

    Early in 2018

    31.

    y=0.00455x+1979.5y=0.00455x+1979.5

    33.

    r=0.999r=0.999


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