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Algebra and Trigonometry

Chapter 4

Algebra and TrigonometryChapter 4
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4.1 Linear Functions

1.

m= 4−3 0−2 = 1 −2 =− 1 2 ; m= 4−3 0−2 = 1 −2 =− 1 2 ; decreasing because m<0. m<0.

2.
m= 1,868−1,442 2,012−2,009 = 426 3 =142 people per year m= 1,868−1,442 2,012−2,009 = 426 3 =142 people per year
3.

y=−7x+3 y=−7x+3

4.

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

5.
6.

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

7.
8.

( 16,0 ) ( 16,0 )

9.
  1. ⓐ f(x)=2x; f(x)=2x;
  2. ⓑ g(x)=− 1 2 x g(x)=− 1 2 x
10.

y=– 1 3 x+6 y=– 1 3 x+6

4.2 Modeling with Linear Functions

1.

ⓐ C( x )=0.25x+25,000 C(x)=0.25x+25,000
ⓑ 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.

ⓐ41,100 ⓑ2020

3.

21.57 miles

4.3 Fitting Linear Models to Data

1.

54°F 54°F

2.

150.871 billion gallons; extrapolation

4.1 Section Exercises

1.

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

3.

d( t )=100−10t d( t )=100−10t

5.

The point of intersection is ( a,a ). ( a,a ). This is because for the horizontal line, all of the y y coordinates are a a and for the vertical line, all of the x x coordinates are a. a. The point of intersection is on both lines and therefore will have these two characteristics.

7.

Yes

9.

Yes

11.

No

13.

Yes

15.

Increasing

17.

Decreasing

19.

Decreasing

21.

Increasing

23.

Decreasing

25.

2

27.

–2

29.

y= 3 5 x−1 y= 3 5 x−1

31.

y=3x−2 y=3x−2

33.

y=− 1 3 x+ 11 3 y=− 1 3 x+ 11 3

35.

y=−1.5x−3 y=−1.5x−3

37.

perpendicular

39.

parallel

41.

f(0)=−(0)+2 f(0)=2 y−int:(0,2) 0=−x+2 x−int:(2,0) f(0)=−(0)+2 f(0)=2 y−int:(0,2) 0=−x+2 x−int:(2,0)

43.

h(0)=3(0)−5 h(0)=−5 y−int:(0,−5) 0=3x−5 x−int:( 5 3 ,0 ) h(0)=3(0)−5 h(0)=−5 y−int:(0,−5) 0=3x−5 x−int:( 5 3 ,0 )

45.

−2x+5y=20 −2(0)+5y=20 5y=20 y=4 y−int:(0,4) −2x+5(0)=20 x=−10 x−int:(−10,0) −2x+5y=20 −2(0)+5y=20 5y=20 y=4 y−int:(0,4) −2x+5(0)=20 x=−10 x−int:(−10,0)

47.

Line 1: m = –10 Line 2: m = –10 Parallel

49.

Line 1: m = –2 Line 2: m = 1 Neither

51.

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

53.

y=3x−3 y=3x−3

55.

y=− 1 3 t+2 y=− 1 3 t+2

57.

0

59.

y=− 5 4 x+5 y=− 5 4 x+5

61.

y=3x−1 y=3x−1

63.

y=−2.5 y=−2.5

65.

F

67.

C

69.

A

71.
73.
75.
77.
79.
81.
83.
85.

y=3 y=3

87.

x=−3 x=−3

89.

Linear, g(x)=−3x+5 g(x)=−3x+5

91.

Linear, f(x)=5x−5 f(x)=5x−5

93.

Linear, g(x)=− 25 2 x+6 g(x)=− 25 2 x+6

95.

Linear, f(x)=10x−24 f(x)=10x−24

97.

f(x)=−58x+17.3 f(x)=−58x+17.3

99.
101.
  1. ⓐ a=11,900, b=1000.1a=11,900, b=1000.1
  2. ⓑ q(p)=1000p –100q(p)=1000p –100
103.
105.

y=− 16 3 y=− 16 3

107.

x=ax=a

109.

y= d c–a x– ad c–a y= d c–a x– ad c–a

111.

y=100x–98y=100x–98

113.

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

115.

$45 per training session.

117.

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.

119.

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

121.

C

4.2 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.

20.01 square units

9.

2,300

11.

64,170

13.

P( t )=75,000+2500t P( t )=75,000+2500t

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 )=12,025−205t C( t )=12,025−205t

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.

2063

31.

y=−2t+180 y=−2t+180

33.

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

35.

y=30t−300 y=30t−300

37.

(10, 0) In the year 1990, the company’s profits were zero

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
  6. ⓕ2,219 people
47.
  1. ⓐC(x) = 0.15x + 10
  2. ⓑThe flat monthly fee is $10 and there is a $0.15 fee for each additional minute used
  3. ⓒ$113.05
49.

P(t) = 190t + 4,360

51.
  1. ⓐ R(t)=−2.1t + 16R(t)=−2.1t + 16
  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.

More than $66,666.67 in sales

4.3 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.

This value of r indicates a strong negative correlation or slope, so C This value of r indicates a strong negative correlation or slope, so C

17.

This value of r indicates a weak negative correlation, so B This value of r indicates a weak negative correlation, so B

19.
21.
23.

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

25.

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

27.

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

29.

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

31.

y=0.121x−38.841,r=0.998 y=0.121x−38.841,r=0.998

33.

(−2,−6),(1,−12),(5,−20),(6,−22),(9,−28); (−2,−6),(1,−12),(5,−20),(6,−22),(9,−28); Yes, the function is a good fit.

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+1985.41 y=0.00587x+1985.41

39.

y=20.25x−671.5 y=20.25x−671.5

41.

y=−10.75x+742.50 y=−10.75x+742.50

Review Exercises

1.

Yes

3.

Increasing

5.

y=−3x+26 y=−3x+26

7.

3

9.

y=2x−2 y=2x−2

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+21 y=−0.2x+21

21.


23.

More than 250

25.

118,000

27.

y=−300x+11,500 y=−300x+11,500

29.
  1. ⓐ 800
  2. ⓑ 100 students per year
  3. ⓒ P( t )=100t+1700 P( t )=100t+1700
31.

18,500

33.

$91,625

35.

Extrapolation

37.


39.

2023

41.

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

43.

2027

45.

7,660

Practice Test

1.

Yes

3.

Increasing

5.

y = −1.5x − 6

7.

y = −2x − 1

9.

No

11.

Perpendicular

13.

(−7, 0); (0, −2)

15.

y = −0.25x + 12

17.

Slope = −1 and y-intercept = 6

19.

150

21.

165,000

23.

y = 875x + 10,625

25.
  1. ⓐ375
  2. ⓑdropped an average of 46.875, or about 47 people per year
  3. ⓒy = −46.875t + 1250
27.
29.

In early 2018

31.

y = 0.00455x + 1979.5

33.

r = 0.999

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