Alg Sem 1 Final Rev
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
too possible sequences that work!” he complains to his mom after school. Do you agree with Michael? Explain
completely.
[ With the initial value and the generator, there is only one sequence fitting this bill: –5, 2, 9, 16, … ]
2. A sequence can be represented by the formula t(n)  5n  2 . What would be a recursive equation for this
same sequence?
[ t(1) = 7, t(n + 1) = t(n) + 5 ]
3. A sequence can be represented by the formula t(1)  6 and t(n 1)  3t(n) . What would be an explicit
equation for this same sequence?
[ t(n) = 2(3^n) ]
4. What is the difference between an explicit equation to represent a sequence and a recursive equation? Be
clear and complete.
[ Answers will vary, but students need to include that a recursive not only tells you had to go from one
term to the next, but also must tell you where to start. Should also include that with an explicit formula,
you can find any term in the sequence directly, but with a recursive formula, you would need the
proceeding terms to find a specific term. ]
5. Write both an explicit equation and a recursive equation for the sequence 7, 10, 13, 16, …
[t(n) = 3n + 4, t(1) = 7 and t(n + 1) = t(n) + 3 ]
6. For each sequence defined recursively below, write the first five terms, and then write an explicit formula for
the sequence.
a.
t(1)  8 and t(n 1)  t(n)  2
b.
t(1) 
c.
t(1)  11 and t(n 1)  t(n)
3
2
and t(n 1)  t(n)  32
[ a: –8, –6, –4, –2, 0, …, t(n) = 2n – 10; b: 3/2, 3, 9/2, 6, 15/2,… t(n) = (3/2)n; c: 11, 11, 11, 11, 11, … t(n)
= 11 ]
7. 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.
[ Sample statements: They begin with $150,210.00 expected profit, which is decreasing at a rate of 18%
per month. At the end of the year, they will have $13,882.42 expected profit. ]
8. For each sequence defined recursively below, write the first five terms, and then write an explicit formula for
the sequence.
a.
t(1)  5 and t(n 1)  101 t(n)
c.
a1  16 and an1  an  3
b.
a1  9 and an1  an  2.5
[ a: 5, 1/2, 1/20, 1/200, 1/2000, …, t(n) = 5(0.1^n); b: 9, 6.5, 4, 1.5, –1,… t(n) = – 2.5n + 11.5; c: –16, –13,
–10, –7, –4, … t(n) = 3n – 19 ]
9. After college, you get a job with a starting salary of $35,500. Your first day on the job is January 1, 2010.
You have a cousin working at the same corporation, but he was hired three years before you. Your contract
states you will receive a pay raise each year of 7%. Suppose your cousin confesses to you that he began at
$40,000 and has been guaranteed a 5% pay raise each year.
a.
Make a table showing the salaries of both you and your cousin for the next five years.
b.
Sketch a graph of both you and your cousin's salaries from now until the time when your earnings catch
up and exceed his earnings. How long will it take before your salary exceeds his?
c.
Write a formula you could use to find: your salary given any number of years and your cousin’s salary
given any number of years. What would an input value of –3 mean?
d.
Suppose that you really want to be making $70,000 a year within five years. At your current position,
what kind of a raise (% increase) would you need in order to get this amount? Does a raise of this size
make sense? What other (legal) ways can you think of that would get you where you want to be
financially within five years at your company?
e.
If your raise were still 7%, but compounded quarterly (given in increments each 3 months), what percent
raise would you expect each quarter? At this rate, what would be your salary in a year and a half?
f.
Which will be greater, your salary in 5 years with a 7% annual raise compounded quarterly (as in part
(e)), or your salary in 5 years with a a 7% annual raise compounded annually (as in part (c))? Justify
your answer.
[ c: If n is the number of years since 2010, your salary is given by
and the cousin’s is
. It would mean the year your cousin was hired. d:
, answers vary;
e:
, where n is the number of quarters, 1.75% each quarter,
; f: compounding quarterly results in a higher salary after 5 years ]
10. Each table below represents an exponential function of the form y  ab x . Copy and complete each table on
your paper and find the corresponding rule. [ a: y  2  5 x , b: y  80(0.5)x ]
a.
b.
x
y
x
y
0
80
0
2
1
40
1
10
2
[20]
2
[50]
3
[10]
3
[250]
4
[5]
4
[1250]
11.
Cleo noticed that two of the equations below correctly fit the table.
x
y
1)
y  3x  1
–4
11
2)
–3
9
–2
7
y  (2x  3)
–1
5
3)
0
3
y  2x  3
1
1
2
–1
4)
y x7
a. Which two fit? Explain how you made your decision.
b. Explain why it makes sense that a table can have two rules?
[ a: Rules (2) and (3); b: When simplified they are the same rule. ]
12. Fill in the blank with >, =, or < to make the statement true: 4  108
4  109 .
[>]
13. Fill in the blank with >, =, or < to make the statement true: 2  10 7
9.9  106 .
[>]
14. Consider the sequence 7, 3, –1, –5, –9, …
a.
What kind of sequence is it? Explain how you know.
b.
Make a table and graph the first six terms of the sequence.
c.
Write a rule t(n) for each term in the sequence, where n is the term number.
y
[ a: Arithmetic. Each term is generated by subtracting 4 from the previous term.
b: See graph at right. c: assuming t(1) = 7, t(n) = –4n + 11 ]
x
15. An arithmetic sequence has a first term of 0 and a 9th term of 12. If 27 is an output of the sequence, which
term number is it? Show and explain how you know you have the correct term number.
[ t(n) = 1.5n; t(18) = 27 ]
16. Choose the correct equation for an arithmetic sequence in which t(4) = 8 and t(10) = 32. Explain your
reasoning or show the work that helped you decide on your answer.
[D]
A.
t(n)  8  32n
B.
C.
t(n)  4n  8
D.
t(n)  32  8n
t(n)  4n  8
17. Consider the sequence 3, 6, 12…
a.
Write the next four terms of the sequence.
b.
What is the generator?
c.
What kind of sequence is it? How can you tell?
d.
Write a rule for this sequence.
[ a: 24, 48, 96, 192; b: multiply by 2, c: geometric, d: assuming t(1) = 3, t(n)  1.5(2)n ]
18. In the game of “Save the Earth” algebraic equations are used to eliminate meteors. If the equation goes
through the coordinate point of the meteor, it is destroyed. Examine the screen shot with meteors on the left and
table of values on the right. Your mission is to eliminate the meteors with the fewest equations possible. Write
down your equations and draw them on the picture.
y
x
7
3
0
–2
–3
y
3
–2
0
4
2
x
[ Two equations will do the job: y   23 x and y   19 x  3 79 . ]
19.
Use tables, rules, and a graph to find and check the solution for the following problem.
Edda has a poodle that weighs 7 pounds and gains 1 pound per year. Walden has a young sheltie that
weighs 2 pounds and gains 1 pound every 6 months. When will the two dogs weigh the same
amount?
[ The dogs will both weigh 12 pounds after 5 years. ]
20. Examine the following systems of equations. Decide for each system which method would be the most
efficient, convenient, and accurate: graphing, substitution, elimination, or the equal values method. Justify your
reasons for choosing one strategy over the others. You do not have to solve the systems.
a.
x = – 2y + 6
x = 3 – 4y
b.
x=4–y
y = 3x + 4
c.
a + b = 10
3a – 4a = 6
[ Answers vary, but generally, (a) is easiest with equal values, (b) with substitution and (c) with
elimination or substitution ]
21. A redwood tree in King’s Canyon National Park, California has a circumference of 220 inches. Tanner
measured the circumference of some other redwoods that had been planted over the last 5 years. His data is
recorded below.
Years since planting
Circumference in inches
3
4
4
5.5
5
7
a. Assuming that the growth rate remains constant, copy and extend the table for the 1st, 2nd, and 6th year.
b. Write a rule for the data in your table.
c. Use your rule to predict the circumference of a tree after 50 years of growth.
d. Use your rule to estimate the age of the tree with a circumference of 220 inches.
[ a: Year 1 = 1 inch; Year 2 = 2.5 inches; Year 6 = 8.5, b: y  1.5x  0.5 , c: y  1.5(50)  0.5 or 74.5
inches , d: 147 years old. ]
22.
Moe and Larry each need to raise $200 to go on a trip with their school band. Moe has $35 in savings
and earns $20 per week delivering papers. Larry has $60 in savings and earns $10 per week doing yard work.
a.
Use at least two different methods to find the time (in weeks) when Moe and Larry will have
the same amount of money in their savings accounts.
b.
How long will each boy have to work to have enough money for the trip?
[ a: After 2.5 weeks, both boys will have $85, b: Moe will have to work 9 weeks, while Larry
will have to work 14 weeks. ]
23. 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
2x  (5  x)  20
x  5  20
x  5  5  20  5
Pearl’s Solution:
x  25
x  25 y  30
[ Yes. The ordered pair at the end does solve both equations. ]
24. Gabby solved this system of equations and found the solution to be (25, -10). Is Gabby correct? How do you
know? Be clear and complete!
y  15 x  15
3x  6y  15
[ Yes, she is. Students can verify this by either solving the system or plugging the point into BOTH
equations and showing that it results in a true statement. ]
25.
Rewrite each of the expressions below with no parentheses and no fractions. Negative exponents are
acceptable in your answer.
[ a:
b: 1.5m6n ]
a.
(2x 5 y 3 )3 (4 xy 4 )2
8x 7 y12
3m 2 n 3
2m(mn)2
b.
26. 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.
+1
-20
–1
5xy
x
5y
x
–4
27. 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
[ a: x  10 , b: y  4x  2 , c: m  3p  2 , d: x 
1
4
y
3
4
]
28. Tinesse is a new student in your study group. She does not understand how to fill in this generic rectangle.
Explain what to do in several complete sentences. [ Answers vary. ]
–1
24x
–3
12xy
0.5y
3
29. Solve each of the equations below for y. For each equation, state the growth and the y-intercept.
a.
y + 3x = 8x – 7
b.
–6x + 5y = –3(2 – y)
c.
x(x + 5) + y = (x + 2)(x – 2)
d.
–2x – 3y + 1 = –x – 4y + 4
[ a: y = 5x – 7, growth = 5, y-intercept = 7; b: y = 3x – 3, growth = 3, y-intercept = –3; c: y = –5x – 4,
growth = –5, y-intercept = –4; d: y = x + 3, growth = 1, y-intercept = 3 ]
30. Write an algebraic equation for each figure below to express the relationship, “Area as a product equals area
as a sum.” [ a: (x  1)(2x  3)  2x 2  x  3 , b: x 2  5x  6  (x  2)(x  3) ]
a.
2x
–3
b.
2x
x
+1
3x
6
31. The multiplication table below has factors along the top row and left column. Their product is where the
row and column intersect. With your team, complete the table with all of the factors and products.
Multiply
(x  5)
(x  3)
2x 2  9x  5
2x 2  8x  6
[ Solution table shown. ]
Multiply
(x  5)
(2x  2)
2x 2  6x  8
(x  3)
(x  4)
(2x  1)
x 2  2x  15
2x 2  8x  6
x 2  9x  20
2x 2  6x  8
2x 2  9x  5
4x 2  6x  2
32.
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.
[ a2 + 2ab + b2 ]
33. Solve the following equation for x. Use the Distributive Property or draw generic rectangles to help you
rewrite the products. [ x = 5/4 ]

(3x  2)(x  1)  3 3x  4  x 2

34. Solve for x. [ x can be any real number ]
(x  6)( x  4)  20  x(x  2)  4
35. Solve for x using any method you prefer. Check your solutions by testing them in the original equation.
a.
|12 – 7x| = 26
b.
c.
–3|5 – 2x| = –12
[ a: x = –2, 38/7; b: x = –4, 10; c: x = 1/2, 9/2 ]
|2(x – 3)| = 14
36. The points (2, 3) and (6, 11) lie on a line. Show your work when answering the following questions.
a. What is the growth rate of this line?
b. What is the y-intercept?
c. What is the equation of the line?
[ a: 2, b: (0, –1), c: y  2x  1 ]
37. What is the equation of each of these lines?
[ a: y = (-1/2)x + 3; b: y = -2x – 6; c: y = (9/4)x – (27/4) ]
38. A new student has joined your study group. Write step-by-step explanations telling her how to use algebra
to find the equation of a line given each of the following. Be sure to include examples.
a) A point and the slope.
b) Two points.
[ Answers vary. ]
39. Without graphing, find the equation of the line that passes through the points (– 4, 3) and (6, –2). Show all
work in a clear and organized manner. [ y   12 x  1 ]
40. Show and explain how to find the equation of the line that passes through the points (– 4, 8) and (5, –1).
[ y  x  4 ]
41. Consider the line containing the points (–1, 3) and (4, 6).
a.
Find the slope of the line.
b.
Find the y-intercept.
c.
Find another point on the line that has integer coordinates. Write the coordinates of the point you found.
d.
Write an equation for the line.
[ a:
3
5
, b:
18
5
, c: Answers will vary; (– 6, 0) is one example, d: y  53 x  18
.]
5
42. For each situation below, write the equation of the situation described.
a.
A line through the points (–12, 3) and (8, 15).
b.
A line with a slope of 4 and a y-intercept of 0.4.
c.
For her birthday, Louise got $100 from her Grammy, but she has been spending $10.00 each week.
d.
A line has intercepts of (-6, 0) and (0, -12).
[ a: y = (3/5)x + (51/5); b: y = 4x + .4; c: y = –10x + 100; d: y = –2x – 12 ]
43. If y  3x  2 is the equation in the web at right, create the
corresponding remaining components of the web. That is, what is the
table? What is the graph? The situation?
y
[ The table and the graph are straight forward, and a portion of the
table is shown. If you expect a more extensive table, then you must
specify how much you require. ]
44. Complete the table below for the rule
x
Graph
.
a. Plot the points and connect them on a complete graph.
b. What does your graph look like? Why does it take this shape?
[ y-values: 13, 8, 5, 4, 5, 8, 13, b: a parabola opening upward ]
Situation
45.
Use your pattern-detection skills to find a rule for the table below.
x
y
2
5
–1
–1
3
7
a.
Copy and complete the table.
b.
Describe how you see the pattern.
c.
What is the rule? Write the rule using words.
d.
How can you tell that your rule is correct?
0
1
–3
–2
[ a: y = –5, x = –1.5; b: Answers vary. c: multiply the input by 2 and then add 1; d: Answers
vary. ]
46.
Mariela graphed all of the equations below but forgot which equation went with which graph. Help her
match each equation with the appropriate graph. Discuss the answers with your group and write a few
sentences explaining how you figured it out.
b
y x3
y
c
d
y  2x  1
y  12 x  1
x
y  2x  3
a
[ Line (a) is y  12 x  1 . Line (b) is y  2x  1 . Line (c) is y  2x  3 .
Line (d) is y  x  3 .]
47. Examine the graphs below and explain what real-world quantities the slope and
y-intercepts represent. Then find the slope and y-intercept for each.
y
a.
y
b.
(0, 12)
(10, 7000)
Weight of a
Bag of Popcorn
(in ounces)
Car
Mileage
(8, 0)
Time
(in minutes)
x
(0, 5000)
x
Days After
the Last Tune-Up
[ a: The slope ( m  23 ) represents the number of ounces of popcorn eaten per minute, and the y-intercept
(b = 12) represents the weight of the bag of popcorn when the eating began; b: The slope ( m  2000
)
10
represents the number of miles traveled per day after the most recent tune-up, and the y-intercept
(b = 5000) represents the car mileage when the tune-up was complete. ]
48. For weeks now, Jeremiah has been collecting data. Every car he comes across, he measures the car’s width
and the car’s length. He graphs this data, and writes the equation of a line he thinks best fits the data. With his
graph and equation complete, he proudly boasts “Give me the width of ANY car and I can tell you, exactly, the
length of that car!”
What do you think? Can Jeremiah give the exact length of the car? Explain completely.
[ This problem is labeled as beginning, but the level can be adjusted based one how much you expect
from your students’ responses. It could even be set up as a Growth Over Time problem. Students should
explain that the equation/line are just a model and therefore not a guarantee of “exactness”. ]
49. Starting in 1960, certain agencies began monitoring life expectancy in the U. S. That first year, the average
woman was expected to live to be 73.23 years old.
After several years, these agencies came up with an equation, y = 0.23x + 73, to give the average lifespan of a
U.S. woman in any year after 1960.
a.
Given this model, determine the life expectancy of an average woman for the next five-decade years:
1970, 1980, 1990, 2000, and 2010?
b.
In the equation, what are the real world interpretations of the slope and y-intercept? Be clear and
complete.
c.
What does the model predict to be the life expectancy of an average woman in the year 2025? How did
you get your answer?
d.
Rate this model: do you think it seems reasonable?
[ a: 1970 – 75.3, 1980 – 77.6, 1990 – 79.9, 2000 – 82.2, 2010 – 84.5; b: The real world interpretation of the
slope is the amount of years added to a woman’s life expectancy each year, the y-intercept is the life
expectancy of a woman in 1960, 73 years old; c: 87.95 years found by plugging x = 65 into the equation;
d: Answers will vary. Seems somewhat reasonable but it implies that life expectancy will be forever
increasing, which probably isn’t likely. ]
50. Does your social networking affect your social life? When analyzing data relating the number of hours a
young person spends on social networking sites, x, researchers found the LSRL, where y is the number of dates
the person went on each month, to be y = 30 – 1.6x. Interpret the slope and y-intercept for this LSRL.
[ The y-intercept is the number of dates a young person would go on in a month if the person were
spending zero hours on social networking sites, while the slope tells us that for each additional hour on a
social networking site, a young person can expect to go on about 1.6 fewer dates that month. ]