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Algebra I (Common Core) - New York Regents August 2014 Exam

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Part I
    Answer all 24 questions in this part. Each correct answer will receive 2 credits. No partial
credit will be allowed. For each statement or question, choose the word or expression that, of those
given, best completes the statement or answers the question. Record your answers on your
separate answer sheet. [48]

1.
1 Which statement is not always true?
  (1) The product of two irrational numbers is irrational.
  (2) The product of two rational numbers is rational.
  (3) The sum of two rational numbers is rational.
  (4) The sum of a rational number and an irrational number is
      irrational.
Answer: 1

2.
2 A satellite television company charges a one-time installation fee and
  a monthly service charge. The total cost is modeled by the function
  y = 40 + 90x. Which statement represents the meaning of each part
  of the function?
  (1) y is the total cost, x is the number of months of service,
      $90 is the installation fee, and $40 is the service charge per
      month.
  (2) y is the total cost, x is the number of months of service,
      $40 is the installation fee, and $90 is the service charge per
      month.
  (3) x is the total cost, y is the number of months of service,
      $40 is the installation fee, and $90 is the service charge per
      month.
  (4) x is the total cost, y is the number of months of service,
      $90 is the installation fee, and $40 is the service charge per
      month.
Answer: 2

3.
3 If 4x2 - 100 = 0, the roots of the equation are
  (1) -25 and 25                 (3) -5 and 5
  (2) -25, only                  (4) -5, only
Answer: 3


4.
                                                                          U
4 Isaiah collects data from two different companies, each with four
  employees. The results of the study, based on each worker’s age and
  salary, are listed in the tables below.
             Company 1                      Company 2
       Worker’s       Salary          Worker’s       Salary
        Age in          in             Age in          in
        Years         Dollars          Years         Dollars
          25          30,000              25          29,000
          27          32,000              28          35,500
          28          35,000              29          37,000
          33          38,000              31          65,000
  Which statement is true about these data?
  (1) The median salaries in both companies are greater than $37,000.
  (2) The mean salary in company 1 is greater than the mean salary in
      company 2.
  (3) The salary range in company 2 is greater than the salary range in
      company 1.
  (4) The mean age of workers at company 1 is greater than the mean
      age of workers at company 2.
Answer: 3

5.
5 Which point is not on the graph represented by y = x2 + 3x - 6?
  (1) (-6,12)                   (3) (2,4)
  (2) (-4,-2)                   (4) (3,-6)
Answer: 4

6.
6 A company produces x units of a product per month, where C(x)
  represents the total cost and R(x) represents the total revenue for the
  month. The functions are modeled by C(x) = 300x + 250 and
  R(x) = -0.5x2 + 800x - 100. The profit is the difference between
  revenue and cost where P(x) = R(x) - C(x). What is the total profit,
  P(x), for the month?
  (1) P(x) = -0.5x2 + 500x - 150
  (2) P(x) = -0.5x2 + 500x - 350
  (3) P(x) = -0.5x2 - 500x + 350
  (4) P(x) = -0.5x2 + 500x + 350
Answer: 2

7.
7 What is one point that lies in the solution set of the system of
  inequalities graphed below?
(1) (7,0)   (3) (0,7)
(2) (3,0)   (4) (-3,5)
Answer: 1

8.
8 The value of the x-intercept for the graph of 4x - 5y = 40 is

Answer: 1

9.
9 Sam and Jeremy have ages that are consecutive odd integers.
  The product of their ages is 783. Which equation could be used to
  find Jeremy’s age, j, if he is the younger man?
  (1) j2 + 2 = 783              (3) j2 + 2j = 783
  (2) j2 - 2 = 783              (4) j2 - 2j = 783
Answer: 3

10.
10 A population that initially has 20 birds approximately doubles every
   10 years. Which graph represents this population growth?

Answer: 3

11.
11 Let f be a function such that f(x) = 2x - 4 is defined on the domain
   2 ≤ x ≤ 6. The range of this function is
   (1) 0 ≤ y ≤ 8                   (3) 2 ≤ y ≤ 6
   (2) 0 ≤ y < ∞                   (4) -∞ < y < ∞
Answer: 1

12.
12 Which situation could be modeled by using a linear function?
   (1) a bank account balance that grows at a rate of 5% per year,
       compounded annually
   (2) a population of bacteria that doubles every 4.5 hours
   (3) the cost of cell phone service that charges a base amount plus
       20 cents per minute
   (4) the concentration of medicine in a person’s body that decays by
       a factor of one-third every hour
Answer: 3

13.
13 Which graph shows a line where each value of y is three more than
   half of x?

Answer: 2

14.
14 The table below shows the average diameter of a pupil in a person’s
   eye as he or she grows older.
                        Age     Average Pupil
                      (years)   Diameter (mm)
                        20           4.7
                        30           4.3
                        40           3.9
                        50           3.5
                        60           3.1
                        70           2.7
                        80           2.3
   What is the average rate of change, in millimeters per year, of a
   person’s pupil diameter from age 20 to age 80?
   (1) 2.4                        (3) -2.4
   (2) 0.04                       (4) -0.04
Answer: 4

15.
15 Which expression is equivalent to x4 - 12x2 + 36?
   (1) (x2 - 6)(x2 - 6)          (3) (6 - x2)(6 + x2)
   (2) (x2 + 6)(x2 + 6)          (4) (x2 + 6)(x2 - 6)
Answer: 1

16.
16 The third term in an arithmetic sequence is 10 and the fifth term is 26.
   If the first term is a1, which is an equation for the nth term of this
   sequence?
   (1) an = 8n + 10                  (3) an = 16n + 10
   (2) an = 8n - 14                  (4) an = 16n - 38
Answer: 2

17.
17 The graph of the equation y = ax2 is shown below.
(1) wider and opens downward
(2) wider and opens upward
(3) narrower and opens downward
(4) narrower and opens upward
Answer: 1

18.
18 The zeros of the function f(x) = (x + 2)2 - 25 are
   (1) -2 and 5                   (3) -5 and 2
   (2) -3 and 7                   (4) -7 and 3
Answer: 4

19.
19 During the 2010 season, football player McGee’s earnings, m, were
   0.005 million dollars more than those of his teammate Fitzpatrick’s
   earnings, f. The two players earned a total of 3.95 million dollars.
   Which system of equations could be used to determine the amount
   each player earned, in millions of dollars?
   (1) m + f = 3.95                (3) f - 3.95 = m
       m + 0.005 = f                   m + 0.005 = f
   (2) m - 3.95 = f                (4) m + f = 3.95
       f + 0.005 = m                   f + 0.005 = m
Answer: 4

20.
      
(1) 4   (3) 8
(2) 6   (4) 11
Answer: 1

21.
21 The table below shows the number of grams of carbohydrates, x, and
   the number of Calories, y, of six different foods.
                   Carbohydrates (x)       Calories (y)
                            8                  120
                           9.5                 138
                           10                  147
                            6                   88
                            7                  108
                            4                   62
   Which equation best represents the line of best fit for this set of data?
   (1) y = 15x                       (3) y = 0.1x - 0.4
   (2) y = 0.07x                     (4) y = 14.1x + 5.8
Answer: 4

22.
22 A function is graphed on the set of axes below.

Answer: 2

23.
23 The function h(t) = -16t2 + 144 represents the height, h(t), in feet, of
   an object from the ground at t seconds after it is dropped. A realistic
   domain for this function is
   (1) -3 ≤ t ≤ 3                    (3) 0 ≤ h(t) ≤ 144
   (2) 0 ≤ t ≤ 3                     (4) all real numbers
Answer: 2

24.
24 If f(1) = 3 and f(n) = -2f(n - 1) + 1, then f(5) =
   (1) -5                         (3) 21
   (2) 11                         (4) 43
Answer: 4


Part II
    Answer all 8 questions in this part. Each correct answer will receive 2 credits. Clearly
indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. For all questions in this part, a correct numerical answer with no work shown will
receive only 1 credit. All answers should be written in pen, except for graphs and drawings,
which should be done in pencil. [16]

25.
25 In the equation x2 + 10x + 24 = (x + a)(x + b), b is an integer. Find algebraically all possible
   values of b.
Answer: MODEL ANSWER GIVEN BELOW
[2] 4 and 6, and correct algebraic work is shown, such as factoring.


26.
26 Rhonda deposited $3000 in an account in the Merrick National Bank, earning 4.2% interest,
   compounded annually. She made no deposits or withdrawals. Write an equation that can be used
   to find B, her account balance after t years.
Answer: MODEL ANSWER GIVEN BELOW
[2] B = 3000(1 + 0.042)t or an equivalent equation in terms of B and t is written.


27.
27 Guy and Jim work at a furniture store. Guy is paid $185 per week plus 3% of his total sales in
   dollars, x, which can be represented by g(x) = 185 + 0.03x. Jim is paid $275 per week plus 2.5%
   of his total sales in dollars, x, which can be represented by f(x) = 275 + 0.025x. Determine the
   value of x, in dollars, that will make their weekly pay the same.
Answer: MODEL ANSWER GIVEN BELOW
[2] 18000, and correct work is shown.


28.
28 Express the product of 2x2 + 7x - 10 and x + 5 in standard form.
Answer: MODEL ANSWER GIVEN BELOW
[2] 2x3 + 17x2 + 25x - 50 and correct work is shown.


29.
29 Let f be the function represented by the graph below.
Determine which function has the larger maximum value. Justify your answer.
Answer: MODEL ANSWER GIVEN BELOW
[2] g(x), and a correct justification is given.


30.
30 Solve the inequality below to determine and state the smallest possible value for x in the solution set.
                                           3(x + 3) ≤ 5x - 3
Answer: MODEL ANSWER GIVEN BELOW
[2] 6, and correct work is shown.


31.
31 The table below represents the residuals for a line of best fit.
                     x        2     3     3     4     6     7    8    9   9   10
                 Residual     2     1    –1    –2    –3    –2    –1   2   0   3
   Plot these residuals on the set of axes below.
Using the plot, assess the fit of the line for these residuals and justify your answer.
Answer: MODEL ANSWER GIVEN BELOW
[2] A correct plot is drawn, poor fit is stated, and a correct justification is written, such as
stating that a pattern is formed.


32.
32 A student was given the equation x2 + 6x - 13 = 0 to solve by completing the square. The first
   step that was written is shown below.
                                            x2 + 6x = 13
   The next step in the student’s process was x2 + 6x + c = 13 + c.
   State the value of c that creates a perfect square trinomial.
Explain how the value of c is determined.
Answer: MODEL ANSWER GIVEN BELOW
[2] 9, and a correct explanation is written.



Part III
    Answer all 4 questions in this part. Each correct answer will receive 4 credits. Clearly
indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs,
charts, etc. For all questions in this part, a correct numerical answer with no work shown will
receive only 1 credit. All answers should be written in pen, except for graphs and drawings,
which should be done in pencil. [16]

33.
33 On the axes below, graph f(x) = |3x|.
If h(x) = f(x - 4), how is the graph of f(x) translated to form the graph of h(x)?
If g(x) = f(x) - 2, how is the graph of f(x) translated to form the graph of g(x)?
Answer: MODEL ANSWER GIVEN BELOW
[4] A correct graph of f(x) is drawn. A correct relationship for g(x) is described, 
such as g(x) is two units below f(x). A correct relationship for h(x) is described, 
such as h(x) is shifted four units to the right of f(x).


34.
The area of a trapezoid is 60 square feet, its height is 6 ft, and one base is 12 ft. Find the
number of feet in the other base.
Answer: MODEL ANSWER GIVEN BELOW
[4] (2A - h b2)/h or an equivalent expression and 8, and correct work is shown.


35.
35 Let f(x) = -2x2 and g(x) = 2x - 4. On the set of axes below, draw the graphs of y = f(x) and
   y = g(x).
Using this graph, determine and state all values of x for which f(x) = g(x).

Answer: MODEL ANSWER GIVEN BELOW
[4] Both functions are graphed correctly, and -2 and 1 are stated.


36.
36 A school is building a rectangular soccer field that has an area of 6000 square yards. The soccer
   field must be 40 yards longer than its width. Determine algebraically the dimensions of the
   soccer field, in yards.
Answer: MODEL ANSWER GIVEN BELOW
[4] 60 and 100, and correct algebraic work is shown.



Part IV
    Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate
the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc.
A correct numerical answer with no work shown will receive only 1 credit. All answers should
be written in pen, except for graphs and drawings, which should be written in pencil. [6]

37.
37 Edith babysits for x hours a week after school at a job that pays $4 an hour. She has accepted a
   job that pays $8 an hour as a library assistant working y hours a week. She will work both jobs.
   She is able to work no more than 15 hours a week, due to school commitments. Edith wants to
   earn at least $80 a week, working a combination of both jobs.
   Write a system of inequalities that can be used to represent the situation.
   Graph these inequalities on the set of axes below.
Determine and state one combination of hours that will allow Edith to earn at least $80 per week
while working no more than 15 hours.
Answer: MODEL ANSWER GIVEN BELOW
[6] x + y ≤ 15 and 4x + 8y ≥ 80 are stated. Both inequalities are graphed and 
shaded correctly with at least one labeled correcty.  A correct combination of
babysitting hours and library hours is stated.


38.
                            High School Math Reference Sheet
1 inch = 2.54 centimeters     1 kilometer = 0.62 mile    1 cup = 8 fluid ounces
1 meter = 39.37 inches        1 pound = 16 ounces        1 pint = 2 cups
1 mile = 5280 feet            1 pound = 0.454 kilogram   1 quart = 2 pints
1 mile = 1760 yards           1 kilogram = 2.2 pounds    1 gallon = 4 quarts
1 mile = 1.609 kilometers     1 ton = 2000 pounds        1 gallon = 3.785 liters
                                                         1 liter = 0.264 gallon
                                                         1 liter = 1000 cubic centimeters

Answer:


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