Algebra 1-2 Flexbook Q1 Solutions – Chapter 2

Chapter 2: Linear Functions

2.1 Write a Function in Slope-Intercept Form

1. 𝑓(−3) = 3; 𝑓(0) = −3; 𝑓(5) = −13

2. 𝑓(−9) = 4; 𝑓(0) = 10; 𝑓(9) = 16

3. 𝑓(𝑥) = 5𝑥 − 3

4. 𝑓(𝑥) = −2𝑥 + 5

5. 𝑓(𝑥) = −7𝑥 + 13

6. 𝑓(𝑥) =1

3𝑥 + 1

7. 𝑓(𝑥) = 4.2𝑥 + 19.7

8. 𝑓(𝑥) = −2𝑥 +5

4

9. 𝑓(𝑥) = −2𝑥

10. 𝑓(𝑥) = −𝑥

11. sample answer: 4 times the sum of a number and 2 is 400

12. −98.8875

13. 12

3℃

14. 40𝑚/𝑚𝑖𝑛

15. 121%

16. 62.52%increase

17. 𝑤 ≈ 6834.78

2.2 Graph a Line in Standard Form

1. 𝑦 = 2𝑥 + 5

2. 𝑦 = −3

8𝑥 + 2

3. 𝑦 = 2𝑥 − 5

4. 𝑦 = −6

5𝑥 − 4

5. 𝑦 = −3

2𝑥 − 4

6. 𝑦 = −1

4𝑥 − 3

7. x-intercept: (6, 0) y-intercept: (0, 4)

8. x-intercept: (−15

2, 0) y-intercept: (0, 6)

9. x-intercept: (8, 0) y-intercept: (0, −4)

10. x-intercept: (−1, 0) y-intercept: (0, −7)

11. x-intercept: (5

2, 0) y-intercept: (0,

3

2)

12. x-intercept: (−7, 0) y-intercept: (0,7

2)

13. x-intercept: 𝑛𝑜𝑛𝑒 y-intercept: (0, 3)

14. sample answer: I think converting to slope intercept form is easier because there are less steps.

15. sample answer: I would graph a vertical line at 𝑥 = −5. There is not y-intercept and the slope is

undefined.

2.3 Horizontal and Vertical Line Graphs

1. 𝑦 = 0

2. 𝑥 = 0

3. E: 𝑥 = 6

4. B: 𝑦 = −2

5. C: 𝑦 = −7

6. A: 𝑦 = 5

7. D: 𝑥 = −4

8.

9.

10.

2.4 Linear Equations in Point-Slope Form

1. 𝑦 − 2 = −1

10(𝑥 − 10)

2. 𝑦 − 125 = −75𝑥

3. 𝑦 + 2 = 10(𝑥 + 8)

4. 𝑦 − 3 = −5(𝑥 + 2)

5. 𝑦 − 12 = −13

5(𝑥 − 10)

6. 𝑦 − 3 = 0

7. 𝑦 + 3 =3

5𝑥

8. 𝑦 − 0.5 = −6𝑥

9. 𝑦 − 7 = −1

5𝑥

10. 𝑦 − 5 = −12(𝑥 + 2)

11. 𝑦 − 5 = −9

10(𝑥 + 7) OR 𝑦 + 4 = −

9

10(𝑥 − 3)

12. 𝑦 − 6 = −𝑥 OR 𝑦 = −1(𝑥 − 6)

13. 𝑦 + 9 = 3(𝑥 + 2)

14. 𝑦 − 32 = −9

5𝑥

15. 𝑦 − 20 =1

40(𝑥 − 100) OR 𝑦 − 25 =

1

40(𝑥 − 300) The length of the spring before it is stretched is

17.5 cm.

16. 𝑦 − 400 = −35

2𝑥 OR 𝑦 − 50 = −

35

2(𝑥 − 20) The depth of the submarine five minutes after it

started surfacing would be 312.5 ft.

2.5 Writing and Comparing Functions

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 𝑑(𝑡) = 1100 − 30𝑡 OR 𝑑(𝑡) = −30𝑡 + 1100

12. 𝑚 = −30

13. 𝑑(𝑡) = 2000 − 20𝑡 OR 𝑑(𝑡) = −20𝑡 + 2000

14. slope: (#11) 𝑚 = −30 (#12) 𝑚 = −20; The distanced traveled each day is larger for the migrating

monarch so it flies at a faster rate.

y-intercepts: (#11) (0, 1100) (#12) (0, 2000); The y-intercept in this scenario represents the total

distance the butterfly must travel, or the amount of miles left to travel on day zero.

x-intercepts: (#11) (362

3, 0) (#12) (100, 0); The x-intercept in this scenario represents the amount of

time it takes to travel the total migrating distance.

Domain: (#11) 0 ≤ 𝑡 ≤ 362

3 (#12) 0 ≤ 𝑡 ≤ 100

Range: (#11) 0 ≤ 𝑑(𝑡) ≤ 1100 (#12) 0 ≤ 𝑑(𝑡) ≤ 2000

15. 𝑓(𝑥) = 1.5 + 3000

16. 𝑚 = 1.5

17. The writer needs to sell 4667 books.

2.6 Applications of Linear Models

1. 𝑦 = 350𝑥 + 1500; x= #of months y=amount paid

Constraints: The number of months (x) would include integers greater than or equal to zero until the car

is paid off. The amount paid would start at $1500 then add an amount of $350 per month until the car

is paid off.

Domain: {0, 1, 2, 3, … } until paid off

Range: {1500, 1850, 2150, … } until paid off

2. 𝑦 =1

2𝑥 +

17

2; x=# of weeks; y= height of the plant (in)

Constraints: The number of weeks could be 0 weeks or greater, including a fraction of a week. The

height could be greater than or equal to 8.5 inches.

Domain: 𝑥 ≥ 0

Range: 𝑦 ≥ 8.5

The height of the rose was 8.5 inches when Anne planted it.

3. 𝑦 =1

40𝑥 + 1; x=weight (lbs.) y=length of spring (m)

Constraints: The weight could be 0 lbs. or greater, including fractions of a pound; the length could be

greater than or equal to 1 m since that is the length of the spring with no weight attached. There would

be a limit to both when the weight would cause the spring to hit the ground.

Domain: 𝑥 ≥ 0

Range: 𝑦 ≥ 1

The spring would be 4.5 meters long when Amardeep hangs from it.

4. 𝑦 =1

2𝑥 + 215; x=weight (lbs.) y=distance stretched (ft.)

Constraints: The weight could be 0 lbs. or greater, including fractions of a pound; the length would be

greater than or equal to 215 ft. which is the length of the cord before it is stretched (within the values

that represent a linear function).

Domain: 𝑥 ≥ 0

Range: 𝑦 ≥ 215

The expected length of the cord would be 290 ft. for a weight of 150 lbs.

5. 𝑦 − 20 =1

40(𝑥 − 100); x=weight (g) y=length (cm)

Constraints: The weight could be 0 g or greater, including fractions of a gram; the length would be

greater than or equal to 17.5 cm which is the length of the cord before it is stretched.

Domain: 𝑥 ≥ 0

Range: 𝑦 ≥ 17.5

6. 𝑦 − 400 = −35

2𝑥; x=time (mins) y=depth (ft)

Constraints: The time would be between 0 and 22.86 minutes (the time it takes to surface) and the

depth would be any measure between 400 and 0 feet.

Domain: [0, 22.86]

Range: [0, 400]

7. 𝑦 − 2500 = 6(𝑥 − 200); x= # of shades sold y=amount $$ made

Constraints: It would be possible to sell zero shades and any whole number greater than zero so the

positive integers are appropriate; the amount made each month would be a minimum of $1300 plus $6

for each shade thereafter.

Domain: positive integers greater than or equal to zero

Range: {1300, 1306, 1312, … }

8. You can only buy one pound of corn.

9. 165 baked fish dinners were sold.

10. Andrew needs to work 36 hours at his $6/hour job to make $366.

11. She needs to invest $2142.86 or less in the account with 7% interest.

12. 𝑦 = 6𝑥 − 16

13. 𝑝 = −19

14. The graph of 𝑥 = 1.5 is a vertical line at 𝑥 = 1.5 where the value of x is always 1.5 for any value of y.

15. No it is not a solution.

16. sample answer: (-4, -2); Quadrant III is (-x,-y)

17. 𝑚 = 0

18.

2.7 Rates of Change

1. Slope is the rate of change when considering a linear equation or function because the rate of change

is constant.

2. traffic light = B; mending tire = E; hills in order of most steep to least steep: A, F, C, D

3. 512

3/ℎ𝑟 OR 155𝑚𝑖/3ℎ𝑟𝑠

4. $24/𝑤𝑒𝑒𝑘

5. sample answer: An elevator moves at 10 ft/sec.

6. x-intercept: (10

3, 0); y-intercept: (0, −2)

7.

8. sample answer:

9. Although this can be graphed as a linear function, keep in mind the constraints of dimes and quarters.

You can’t have a negative amount of either and you can’t have a portion of either (i.e. 37.5 dimes). In

reality, the graph should be a set of discrete points rather than a continuous linear function.

10. domain: {−2, −1, 0, 1, 2}; range: {2, 1, 0, 1, 2}

11. 𝑦 = 6.75

12. 3𝑥 + 1 = 2𝑥 − 35

−1 − 1 subtraction property of equality

3𝑥 = 2𝑥 − 36 substitution property of equality (simplify)

−2𝑥 − 2𝑥 subtraction property of equality

𝑥 = −36 substitution property of equality (simplify)

13. 𝑎 = 3

Quick Quiz

1. x-intercept: (25

3, 0); y-intercept: (0,

35

36)

2. 𝑚 = −1

13

3.

4.

5. sample answer: Membership has been steadily increasing over the last 10 years. The increase in

membership was the same from year to year for the first two years.

2.8 Linear and Non-Linear Function Distinction

1. non-linear

2. linear

3. linear

4. linear

5. non-linear

6. linear

7. linear

8. non-linear

9. linear

10. non-linear

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

2.9 Comparing Function Models

2.29 linear

2.30 non-linear

2.31 linear

2.32 not quadratic

2.33 quadratic

2.34 not quadratic

2.35 not exponential

2.36 exponential

2.37 exponential

2.38 exponential

2.39 linear

2.40 quadratic