Precalculus 2e

Chapter 2

2.1Linear 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 .

3 .

$y−2=−2( x+2 ) y−2=−2( x+2 )$ ; $y=−2x−2 y=−2x−2$

4 .

$y−0=−3( x−0 ) y−0=−3( x−0 )$; $y=−3x y=−3x$

5 .

$y=−7x+3 y=−7x+3$

6 .

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

2.2Graphs of Linear Functions

1 .
2 .

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

3 .
4 .

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

2.3Modeling 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 2.4Fitting 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 . $d( t )=100−10t d( t )=100−10t$ 7 . Yes. 9 . No. 11 . No. 13 . No. 15 . Increasing. 17 . Decreasing. 19 . Decreasing. 21 . Increasing. 23 . Decreasing. 25 . 3 27 . $–13–13$ 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 . $−54−54$ 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, $f(x)=5x−5f(x)=5x−5$ 51 . Linear, $g(x)=−252x+6g(x)=−252x+6$ 53 . Linear, $f(x)=10x−24f(x)=10x−24$ 55 . $f(x)=−58x+17.3 f(x)=−58x+17.3$ 57 . 59 . a. $a=11,900a=11,900$; $b=1000.1b=1000.1$ b. $q(p)=1000p−100q(p)=1000p−100$ 61 . 63 . $x=−163x=−163$ 65 . $x=ax=a$ 67 . $y=dc−ax−adc−ay=dc−ax−adc−a$ 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 .

;

15 .

;

17 .

;

19 .

$Neither Neither$

21 .

$Perpendicular Perpendicular$

23 .

$Parallel Parallel$

25 .

$g(x)=3x−3g(x)=3x−3$

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 .
41 .
43 .
45 .
47 .
49 .
51 .
53 .
55 .
57 .
59 .
1. $g(x)=0.75x−5.5g(x)=0.75x−5.5$
2. 0.75
3. $(0,−5.5)(0,−5.5)$
61 .

$y=3y=3$

63 .

$x=−3x=−3$

65 .

no point of intersection

67 .

69 .

71 .

$y=100x−98 y=100x−98$

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. : 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 .

: 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 .

$y=30t−300 y=30t−300$

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. $R( t )=16−2.1t R( t )=16−2.1t$
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 .

15 .

C

17 .

B

19 .
21 .
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 .

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

33 .

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

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 .

$y=20.25x−671.5y=20.25x−671.5$

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 .

$y=2x−2y=2x−2$

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 .

$y=−0.2x+21y=−0.2x+21$

21 .
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.

37 .
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 .

$y=−1.5x−6y=−1.5x−6$

7 .

$y=−2x−1y=−2x−1$

9 .

No.

11 .

Perpendicular

13 .

$(−7,0)(−7,0)$; $(0,−2)(0,−2)$

15 .

$y=−0.25x+12y=−0.25x+12$

17 .
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 .
29 .

Early in 2018

31 .

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

33 .

$r=0.999r=0.999$