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Algebra I (Common Core) - New York Regents Fall 2013 Exam Sample

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1.
1   Which ordered pair is not in the solution set of
(1)   (5,3)
(2)   (4,3)
(3)   (3,4)
(4)   (4,4)
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.REI.3 because it represents a system of inequalities
            where students need to determine viable solutions and a nonviable solution.
Rationale: Option 2 is correct.
A student can also determine the answer to the question using substitution.


2.
2   If the quadratic formula is used to find the roots of the equation x2 − 6x − 19 = 0, the
    correct roots are
    (1) 3 ± 2 √7                              (3)     3 ± 4 √14
    (2) −3 ± 2 √7                             (4)     −3 ± 4 √14
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.REI.4b and its NYS clarification (solutions may
            include simplifying radicals) because the student must solve a quadratic
            equation and understand the process of simplifying radicals.
Rationale: Option 1 is correct.


3.
3   Which statistic would indicate that a linear function would not be a good fit to model a data set?
(1) r = -0.93                                (2) r = 1

Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to S.ID.6c and its corresponding clarification (both
            correlation coefficient and residuals will be addressed in this standard)
            because the student must determine the fit of the data by interpreting a
            correlation coefficient and residual plots.
Rationale: Option 3 is correct. A correlation coefficient close to –1 or 1 indicates a good fit.
           For a residual plot, there should be no observable pattern and a similar distribution
           of residuals above and below the x-axis.



4.
4   On the set of axes below, graph the function represented by y = 3√(x − 2) for the domain
    −6 ≤ x ≤ 10.

Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to F.IF.7b because the students will graph a cube root function.
Rationale:
The graph must be drawn for the given domain only. The graph must not include arrows.
Rubric:
[2] A correct graph is drawn for the given interval.


5.
5   Solve 8m2 + 20m = 12 for m by factoring.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.SSE.3a and its NYS clarification (includes trinomials
            with leading coefficients other than 1) because it requires the student to factor a
            quadratic with a leading coefficient other than 1.
Rationale:
                 8m2 + 20m = 12
                 8m2 + 20m − 12 = 0
                 4(2m2 + 5m − 3) = 0
                 4(m + 3)(2m − 1) = 0
                 m+3= 0        2m − 1 = 0
                 m = −3       2m = 1
                                m= 1/2
                      m = −3 and 1/2
                  
Rubric:
      
[2] –3 and  1/2, and correct work is shown.


6.
6   Ryker is given the graph of the function y = ½ x2 – 4. He wants to find the zeros of the function,
                                                  
    but is unable to read them exactly from the graph.
Find the zeros in simplest radical form.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This item aligns to A.REI.4 and its clarification (solutions may include
            simplifying radicals) because it requires a student to choose an appropriate
            method of solving a quadratic function, and the solutions are simplified in
            radical form.
Rationale:
Rubric:
[2] ±2 √2 , and correct work is shown.


7.
7   Emma recently purchased a new car. She decided to keep track of how many gallons of gas
    she used on five of her business trips. The results are shown in the table below.
                                Miles Driven      Number of
                                                  Gallons Used
                                     150                7
                                     200               10
                                     400               19
                                     600               29
                                    1000               51
    Write the linear regression equation for these data where miles driven is the independent
    variable. (Round all values to the nearest hundredth.)
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to S.ID.6a and its corresponding clarification (includes the
            use of the regression capabilities of the calculator) because it requires the student
            to write a linear regression equation for the given data while using the regression
            capabilities of the calculator.
Rationale: The linear regression equation, y = 0.05x − 0.92, was found using the regression
           capabilities of the calculator.
Rubric:
[2] The equation y = 0.05x − 0.92 or an equivalent equation is written.


8.
8   Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of
    three mints contains a total of 10 Calories.
    On the axes below, graph the function, C, where C (x) represents the number of Calories in x mints.
Write an equation that represents C (x).
A full box of mints contains 180 Calories. Use the equation to determine the total number of
mints in the box.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns with A.CED.3 because students must write an equation,
            then use the equation to determine a viable solution.
Rationale:
                               
The slope of the given line is  10/3      , therefore the equation representing C (x) is
 C (x) =   (10/3) x
 180 = (10/3)x
 540 = 10 x
 54 = x
There are 54 mints in the box.
Rubric:
[4] A correct graph is drawn, C (x) = (10/3) x, y = (10/3) x or an equivalent equation is written,
    and correct work using the equation is shown to find 54.


9.
9   David has two jobs. He earns $8 per hour babysitting his neighbor’s children and he earns
    $11 per hour working at the coffee shop.
    Write an inequality to represent the number of hours, x, babysitting and the number of hours, y,
    working at the coffee shop that David will need to work to earn a minimum of $200.
David worked 15 hours at the coffee shop. Use the inequality to find the number of full hours he
must babysit to reach his goal of $200.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.CED.3 because the student must write an inequality
            and interpret solutions as viable or nonviable options.
Rationale: 8x + 11y ≥ 200
             8x + 11(15) ≥ 200
             8x + 165 ≥ 200
             8x ≥ 35
             x ≥ 4.375
              5 hours
Rubric:
[4] 8x + 11y ≥ 200 or equivalent, 5 and correct work is shown.


10.
10 On the set of axes below, graph the function y = |x + 1| .
State the range of the function.
State the domain over which the function is increasing.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to F.IF.7b because students must draw the graph of an
            absolute value function.
The graph should be drawn to include the decreasing and increasing portions of the graph. Since a
domain is not stated, arrows must be included on the graph of the function.
The range of the function is y ≥ 0, [0, ∞), {y | y ≥ 0, where y is a rational number}, or all real numbers ≥ 0.
The function is increasing for x > − 1, (−1,∞), {x | x > − 1, where x is a real number}, or all real
numbers > –1.
Rubric:
[4] A correct graph is drawn and range, y ≥ 0 and domain, x > –1 or equivalent intervals are
    stated.


11.
11 The table below lists the total cost for parking for a period of time on a street in Albany, N.Y. The
   total cost is for any length of time up to and including the hours parked. For example, parking for
   up to and including 1 hour would cost $1.25; parking for 3.5 hours would cost $5.75.
                                      Hours           Total
                                      Parked          Cost
                                         1            1.25
                                         2            2.50
                                         3            4.00
                                         4            5.75
                                         5            7.75
                                         6            10.00
    Graph the step function that represents the cost for the number of hours parked.
Explain how the cost per hour to park changes over the six-hour period.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to F.IF.7b because it requires a student to graph a step
            function. Students may interpret the cost per hour of the function from the table
            or the graph.
Rationale:
The cost for each additional hour increases after the first 2 hours. This can be determined by
viewing the increasing gaps between the steps or by calculating the cost for each additional
hour after two hours.
Rubric:
[4] A correct graph is drawn and a correct explanation is stated.


12.
12 At an office supply store, if a customer purchases fewer than 10 pencils, the cost of each pencil is
   $1.75. If a customer purchases 10 or more pencils, the cost of each pencil is $1.25.
    Let c be a function for which c (x) is the cost of purchasing x pencils, where x is a whole number.
              ⎧1.75x, if 0 ≤ x ≤ 9
     c ( x) = ⎨
              ⎩1.25x, if x ≥ 10
    Create a graph of c on the axes below.
A customer brings 8 pencils to the cashier. The cashier suggests that the total cost to purchase
10 pencils would be less expensive. State whether the cashier is correct or incorrect. Justify your
answer.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to F.IF.7b because a student has to graph a piecewise-
            defined function.
Rationale:
The data points are not connected because the points represent discrete data and the values for
the data can be included on the axes.
Data points can be connected or drawn as a straight line if plotting each individual data point
would represent a line.
Since 8 pencils cost $14 and 10 pencils cost $12.50, the cashier is correct. The student can
calculate the cost of pencils or use the graph to identify that 10 pencils are cheaper than 8 pencils.
Rubric:
[4] A correct graph is drawn and the cashier is correct or an equivalent answer is stated and a
    correct justification is written.


13.
13 About a year ago, Joey watched an online video of a band and noticed that it had been viewed
   only 843 times. One month later, Joey noticed that the band’s video had 1708 views. Joey made
   the table below to keep track of the cumulative number of views the video was getting online.
                 Months Since First Viewing               Total Views
                               0                               843
                               1                              1708
                               2                         forgot to record
                               3                              7124
                               4                             14,684
                               5                             29,787
                               6                             62,381
    a) Write a regression equation that best models these data. Round all values to the nearest
       hundredth. Justify your choice of regression equation.
b) As shown in the table, Joey forgot to record the number of views after the second month.
   Use the equation from part a to estimate the number of full views of the online video that
   Joey forgot to record.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to S.ID.6a and its corresponding clarification (includes the
            use of the regression capabilities of the calculator) because students will use the
            regression capabilities of their calculator to determine the exponential regression
            equation, and solve a problem using the equation created within the context of
            the data.
                                             
Rationale:     The exponential regression equation, y = (836.47)(2.05)x , was found using the
               regression capabilities of the calculator.
               The student chose the exponential regression because the data appear to increase
               at an exponential rate. A scatter plot of the data supports an exponential model.
               For the second month:
                  y = (836.47)(2.05)x
                  y = 3515
Rubric:
[4] y = 836.47 (2.05)x , a correct justification, and 3515 are written.


14.
14 Use the data below to write the regression equation (y = ax + b) for the raw test score based on
   the hours tutored. Round all values to the nearest hundredth.
                          Tutor       Raw Test           Residual
                         Hours, x      Score        (Actual – Predicted)
                             1            30                 1.3
                             2            37                 1.9
                             3            35                –6.4
                             4            47                –0.7
                             5            56                 2.0
                             6            67                 6.6
                             7            62                –4.7
    Equation: _______
    Create a residual plot on the axes below, using the residual scores in the table above.
Based on the residual plot, state whether the equation is a good fit for the data. Justify your
answer.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to S.ID.6b because the residuals were plotted and analyzed
            to assess the fit of the function. Students may have to calculate the residual using
            a calculator to create a residual plot.
Rationale: The regression equation y = 6.32x + 22.43 was found using the regression
           capabilities of the calculator.
Based on the residual plot, the equation is a good fit for the data because the residual values are
scattered without a pattern and are fairly evenly distributed above and below the x-axis.
Rubric:
[4] y = 6.32 x + 22.43, a correct residual graph is drawn, and a correct justification is written.


15.
15 A local business was looking to hire a landscaper to work on their property. They narrowed
   their choices to two companies. Flourish Landscaping Company charges a flat rate of $120
   per hour. Green Thumb Landscapers charges $70 per hour plus a $1600 equipment fee.
    Write a system of equations representing how much each company charges.
    Determine and state the number of hours that must be worked for the cost of each company to
    be the same. [The use of the grid below is optional.]
    If it is estimated to take at least 35 hours to complete the job, which company will be less
    expensive? Justify your answer.
Answer: MODEL ANSWER GIVEN BELOW
Commentary: The item aligns to A.REI.6 because it requires a student to write and solve
            a system of linear equations in two variables.
Rationale :     Flourish Landscaping Company
                      y = 120 x
                Green Thumb Landscapers
                      y = 70 x + 1600
                120 x = 70 x + 1600
                −70 x − 70 x
                50 x / 50 = 1600 / 50
 
                x = 32 hours
                 y = 120 ( 35 ) = $4200
                 y = 70 ( 35 ) + 1600 = $4050
                Green Thumb Landscapers would be less expensive.
Additionally, as shown on the next page, a grid can be used to estimate the intersection of the
two lines, while the exact point can be calculated. The graph also shows that Green Thumb
Landscapers would be less expensive if the job takes at least 35 hours to complete.
Rubric:
[6] y = 120 x and y = 70 x + 1600 or equivalent, 32, and Green Thumb Landscapers with a
    correct justification written.



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