Algebra Semester One Review 1. Michael’s teacher gave him an assignment: “Use an initial term of –5 and a generator of ‘+ 7’ to create an arithmetic sequence,” she told the class. Michael does not think this is a good assignment because “there are two possible sequences that work!” he complains to his mom after school. Do you agree with Michael? Explain completely.

2. Simplify using only positive exponents.

a. b. c. 3. Write an explicit equation for each sequence below.

a. 3, 1, –1, … b. −5, −13, −21, … c. 7

4,

5

2,

13

4, 4,

19

4 … d. –2, –5, –8, –11, …

e. 1, −5, 25, −125, … f. −3, 9, −27, 81, … g. −0.2, 1, −5, 25, … 4. What is the difference between an explicit equation to represent a sequence and a recursive equation? Be clear and complete.

5. Find the coordinates of the x- and y-intercepts of the given line: 1553 yx

6. Antonio is president of Maths-R-Us and has asked his financial advisor for a report about the company’s expected profits. The advisor told him that the profits were expected to follow the equation

y 150210(0.82)x over the next 12 months. Tell Antonio everything you can tell about the company from the

equation. 7. Write system of equations to solve each problem. a. You are selling tickets at a high school football game. Student tickets cost $2 and general admission tickets cost $3. You sell 1957 tickets and collect $5035. How many of each type of ticket did you sell? b. A retailer offers two options for satellite TV service. A customer may buy the dish for $150 and then pay $25 per month for service, or simply pay $35 month for service and dish rental. How many months will it take for the two plans to cost the same amount? If you choose the dish-rental option and keep the service for 24 months, will you save money? How much? c. Ronnie has 57 nickels and quarters. If Ronnie has $7.25, how many nickels and quarters does he have? 8. Each table below represents an exponential function of the form y abx . Find the corresponding rule. a. b.

x y

0 80

1 40

2 [20]

3 [10]

4 [5]

x y

0 2

1 10

2 [50]

3 [250]

4 [1250]

9. Solve the inequality . Write your answer as a compound inequality.

10. Solve the inequality |6 − 4𝑥| ≥ 30. Write your answer as a compound inequality. 11. Draw a graph of the solution region for each system of inequalities below.

a. {𝑦 > 𝑥 − 4

𝑦 ≤ −5

4𝑥 + 6

b. {−2𝑥 +

1

2𝑦 ≤

1

2

3𝑦 + 6 > 3𝑥

12. Write an equation in slope-intercept form for the line with the given condition that contains the point (6, –2). a. parallel to the line 52 yx ____________________

b. perpendicular to the line 43 xy __________________

13. Solve and graph each inequality. Determine if the given point is a solution.

12 xy (3, 4) 2. 623 yx (0, 0) 3. 3𝑥 + 18𝑦 ≤ 0 (0, 13)

14. Use each pair of points to write the equation of a line in point-slope form, and then write the equation in slope-intercept form. a. (5, –2), (–2, 5) b. (–3, 3), (– 4, 4)

15. Find the domain and range of each function:

16. The ordered pairs in the sets shown below are of the form (x, y). In which set(s) is y a function of x?

A. {(1, 3), (2, 6), (3, 1), (6, 3)}

B. {(1, 3), (3, 1), (3, 4), (4, 3)}

C. {(1, -2), (1, 0), (1, 5), (1, 7)}

D. {(0, 3), (1, 4), (3, 4), (2, 8)}

17. a. If 42)( xxf , find x when the f(x) = 12. b. If xxxf 2)( , find the f(x) when x = – 6.

18. Does the table represent a function? If yes, give the domain and range.

a. b.

19. Write the equation of the line that contains the points (2, 3) and (6, 11). 20. Pearl solved a system of equations using substitution. Did she do it correctly? How do you know? If she did not, find her error and solve the system correctly.

System: y 5 x

2x y 20 Pearl’s Solution:

2x (5 x) 20

x 5 20

x 5 5 20 5

x 25

x 25 y 30

21. Rewrite each of the expressions below with no parentheses and no fractions or negative exponents.

a. (2x5y3 )3(4xy4 )2

8x7y12 b. 3m2n3

2m(mn)2

22. Find the missing dimensions (length and width) or area of each part and write the area of the rectangle as a product and a sum. a. b.

23. Solve the following equations for the indicated variable. Show all of your work.

a. Solve for x: 2(x 1) x 12 b. Solve for y: 8x 2y 4

c. Solve for m: 4 p 4 2(m p) d. Solve for x: y 4x 3

24. Solve and graph each inequality. a. −4 < 𝑥 − 2 < 3 b. −3 < −2𝑥 + 1 ≤ 9

c. 2𝑥 + 6 < −2 𝑜𝑟 4𝑥 − 5 > 3 d. −14 > 𝑤 + 3 𝑜𝑟 3𝑤 ≥ −27

25. Solve.

a. 6𝑥 = 1296 b. 9𝑥 = 3𝑥+2 c. 42𝑥 = 8𝑥−2

26. Multiply each using a generic rectangle.

X Y

-2 4

-1 1

0 -2

2 -8

4 -14

X Y

-2 -1

-1 0

0 1

0 2

2 3

+1

x

5y –4

–1

5xy

x

-20

a. (𝑥 − 5)(𝑥 − 1) b. (2𝑥 + 3)(𝑥 − 3) c. (2𝑥 − 1)(5𝑥 − 4) d. (𝑥 + 3)(𝑥 − 7) 27. On a recent online math quiz, Leonhard faced the question: “True or false: (a + b)2 = a2 + b2.” Leonhard quickly typed in “false”, and the screen promptly showed “Congratulations! You are correct! So if it doesn’t equal a2 + b2, what does it equal?”

Now Leonhard was stumped. Help him out: what does it equal, and how do you know? Be clear so Leonhard can understand this question.

28. Solve for x using any method you prefer. Check your solutions by testing them in the original equation. a. |𝑥 − 7| = |2𝑥 − 2| b.

|2(x – 3)| = 14 c. –3|5 – 2x| = –12 d. |4𝑥 + 3| = 3 − 𝑥

29. Simplify leaving in simplest radical form.

a. √864 b. 36√160

−3√8 c. (−5√10)(−6√15)

30. A population of 500 turtles is growing each month at a rate of 7.5%.

a) Write an equation that expresses the number of turtles at time x. b) About how many turtles will there be in 6 months?

31. Your new car cost $16000 but it depreciates in value by about 2% each month. a) Write an equation that would indicate the value of the car at x months. b) How much will your car be worth in 2 years?

32. A town currently has a population of 15,000 but decreases in size by 4.2% each year.

a) Write an equation that expresses the number of people living in the town at time x. b) About how many people will be living there in 20 years? 33. Your new computer cost $1500 but after each year it is only worth 2/3 of the value from the previous year.

a). Write an equation that would indicate the value of the computer at x years. b). How much will your computer be worth in 4 years?

34. You invest $12,000 in an account with 3% annual interest, compounded quarterly. Assume you don’t touch the money or add money other than the earned interest.

a) Write an equation that gives the amount of money, y, in the account after x quarters. b) How much money will you have in the account after 10 years?

35. You invest $10,000 in an account with 5.2% interest, compounded annually. Assume you don’t touch the money or add money other than the earned interest.

a) Write an equation that gives the amount of money, y, in the account after x years. b) How much money will you have in the account after 5 years? c) If you would have invested your $10,000 in a different account instead that offered 6% annual interest

compounded monthly, how much money would you have in the account after 5 years?

36. The table shows the number of people x in a room and the temperature in the room in degrees Fahrenheit, y.

The results are shown at right.

a. Identify and interpret the correlation coefficient.

b. Interpret the slope and y-intercept of the line of best fit.

c. Approximate the temperature when 15 people are in the room.

37. The table shows the average number of minutes y per kilometer for runners and the total distance of a running race, x (in kilometers).

The results are shown at right.

a. Identify and interpret the correlation coefficient.

b. Interpret the slope and y-intercept of the line of best fit.

c. Approximate the average number of minutes per kilometer when the distance of a race is 31 kilometers.

Algebra Semester 1 Final Review Answers:

1. There is only one arithmetic sequence with this first term and common difference

2. a. 9x2 b. 1

𝑥 c.

1

25

3. 18. a. 𝑎𝑛 = 3 + (𝑛 − 1)(−2) b. 𝑎𝑛 = −5 + (𝑛 − 1)(−8) c. 𝑎𝑛 =7

4+ (𝑛 − 1)

3

4

d. 𝑎𝑛 = −2 + (𝑛 − 1)(−3) e. 𝑎𝑛 = (−5)𝑛−1 f. 𝑎𝑛 = −3(−3)𝑛−1 g. 𝑎𝑛 =−0.2(−5)𝑛−1

4. Answers will vary, but you need to include that a recursive equation only tells you how to go from one

term to the next, and must tell you where to start. Should also include that with an explicit formula, you

can find any term in the sequence directly.

5. (−5,0) and (0, −3) 6. His original profit was $150,210, but is decreasing by 18%

x 0 1 2 3 4 5 6 7 8

y 76 76 77 77 78 79 79 80 82

LinReg

𝑦 = 𝑎 + 𝑏𝑥 a = 75.42222222

b = .7

r2 = .9316901408

r = .9652409755

x 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9

y 5.4 5.6 5.7 5.9 6.0 6.1 6.3 6.5 6.9

LinReg

𝑦 = 𝑎 + 𝑏𝑥 a = 5.2027777778

b = .0543010753

r2 = .9647856242

r = .9822350148

7. a. 𝑆+𝐺=1957

2𝑆+3𝐺=5035, 836 students, 1121 general admission

b. 𝐶=150+25𝑚

𝐶=35𝑚 The cost to rent over 24 months is $840. The cost to buy is $750, so it is $90 less

c. 𝑛+𝑞=57

.05𝑛+.25𝑞=7.25 ; he has 22 quarters and 35 nickels

8. a. 𝑦 = 2(5)𝑥 b. 𝑦 = 80 (1

2) ^𝑥

9. –10 < x < 5 10. 𝑥 ≤ −6 or 𝑥 ≥ 9

11.

12. a. 𝑦 = −2𝑥 + 10 b. 𝑦 =1

3𝑥 − 4

13.

14. a. {

𝑦 − 5 = −(𝑥 + 2)𝑦 + 2 = −(𝑥 − 5)

𝑦 = −𝑥 + 3 b. {

𝑦 − 4 = −(𝑥 + 4)𝑦 − 3 = −(𝑥 + 3)

𝑦 = −𝑥

15. a. D: all real #s b. D: 𝑥 ≥ 0 c. D: -2, 1, 2, 4

R: 𝑦 ≤ 2 R: 𝑦 ≤ 4 R: -2, 1, 3

16. a. yes b. no c. no d. yes 17. a. x = -4 b. f(x) = 42

18. a. yes. D: -2, -1, 0, 2, 4 R: -14, -8, -2, 1, 4 b. no

19. 𝑦 = 2𝑥 − 1

20. Yes, she solved correctly. I substituted the solution into both equations.

21. a. 16𝑥10𝑦5 b. −3𝑛

2𝑚5

22. a. (5𝑦 − 1)(𝑥 + 20) = −𝑥 + 5𝑥𝑦 + 100𝑦 − 20 b. (5𝑦 − 4)(𝑥 + 1) = 5𝑥𝑦 + 5𝑦 − 4𝑥 − 4

23. a. x = 10 b. 𝑦 = −4𝑥 + 2 c. 𝑚 = 3𝑝 − 2 d. 𝑥 =𝑦−3

4𝑜𝑟

1

4𝑦 −

3

4

24. a. −2 < 𝑥 < 5 b. −4 ≤ 𝑥 < 2 c. 𝑥 < −4 𝑜𝑟 𝑥 > 2 d. 𝑤 < −17 𝑜𝑟 𝑤 ≥ −9

25. a. x = 4 b. x = 2 c. x = -6

26. a. 𝑥2 − 6𝑥 + 5 b. 2𝑥2 − 3𝑥 − 9 c. 10𝑥2 − 13𝑥 + 4 d. 𝑥2 − 4𝑥 − 21

27. (𝑎 + 𝑏)(𝑎 + 𝑏) = 𝑎2 + 2𝑎𝑏 + 𝑏2

28. a. 𝑥 = −5, 3 b. 𝑥 = 10, −4 c. 𝑥 =1

2,

9

2 d. 𝑥 = −2, 0

29. a. 12√6 b. −24√5 c. 150√6

30. 𝑦=500(1.075)𝑥

≈771 31.

𝑦=16000(.98)𝑥

$9852.49 32. 𝑦 = 15000(.958)𝑥; ≈ 6359

33.𝑦=1500(

2

3)^𝑥

$296.30 34.

𝑦=12000(1.0075)4𝑡

$16180.18 35. {

𝑦 = 10000(1.052)𝑥

$12884.83𝑦 = 10000(1.005)60; $13488.50

36. a. 0.9652 shows a strong positive correlation b. y-intercept is 75.4222; when there are no people in the

room the temperature is about 75.4°. Slope is .7; for every additional person in the room the

temperature increases by .7° c. ≈ 85.92°

37. a. .98223 shows a strong positive correlation b. y-intercept is 5.2; when you run 0 km, you average 5.2

min/km – this does not make sense in the context of this problem. Slope is .054; each 1 km increase in

distance adds .054 to the average run time. c. f(31) = 6.874 min/km