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

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1.
1. Given the functions g(x), f(x), and h(x) shown below:
The correct list of functions ordered from greatest to least by average rate of change over the
interval 0 ≤ x ≤ 3 is
(1)   f(x), g(x), h(x)
(2)   h(x), g(x), f(x)
(3)   g(x), f(x), h(x)
(4)   h(x), f(x), g(x)
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to F.IF.6 because it assesses a student’s ability to calculate
            the average rate of change of a function presented symbolically, as a table, and
            graphically.
Rationale:
              Option 4 is correct. Over the interval 0 ≤ x ≤ 3, the average rate of change for


2.
2. The graphs below represent functions defined by polynomials. For which function are the zeros
   of the polynomials 2 and –3?

Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.APR.3 because it requires a student to identify the
            graph of a polynomial with two given zeros.
Rationale: Option 3 is correct. The graph of the polynomial intersects the x-axis at points
           (–3, 0) and (2, 0). These are the only points on the graph where y = 0.


3.
3. For which function defined by a polynomial are the zeros of the polynomial –4 and –6?
       (1)    y =  x2 − 10 x − 24
       (2)    y =  x2 + 10 x + 24
       (3)    y =  x2 + 10 x − 24
       (4)    y =  x2 − 10 x + 24
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This question aligns to A.APR.3 because it requires a student to identify the
            equation of a polynomial with two given zeros.
Rationale: Option 2 is correct.
            x = –4 and x = –6
            x + 4 = 0 and x + 6 = 0
            0 = (x + 4)(x + 6)
            0 = x2 + 4x + 6x + 24
            0 = x2 + 10x + 24



4.
4. The length of the shortest side of a right triangle is 8 inches. The lengths of the other two sides
   are represented by consecutive odd integers. Which equation could be used to find the lengths
   of the other sides of the triangle?
    (1)   82 + (x + 1) = x2
    (2)   x2 + 82 = (x + 1)2
    (3)   82 + (x + 2)2 = x2
    (4)   x2 + 82 = (x + 2)2
Answer: MODEL ANSWER GIVEN BELOW
Commentary: This item aligns to A.CED.1 because the student creates an equation in one
            variable that can be used to solve a problem.
Rationale: Option 4 is correct.


5.
5. Donna wants to make trail mix made up of almonds, walnuts and raisins. She wants to mix one
   part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts
   cost $9 per pound, and raisins cost $5 per pound.
   Donna has $15 to spend on the trail mix. Determine how many pounds of trail mix she can
   make. [Only an algebraic solution can receive full credit.]
Answer: MODEL ANSWER GIVEN BELOW
Key: 2 pounds of trail mix
Commentary: This question aligns to A.CED.1 because the student creates equations in one
            variable and uses them to solve a problem.
Rationale: Let x = pounds of an ingredient. Then the number of pounds of trail mix is
           represented by the expression x + 2x + 3x. Therefore, the number of pounds of trail
           mix is 6x. Then,
              12x + 9(2x) + 5(3x) = 15
                              45x = 15
                                x = 1/3
                                      
           So, 6 (1/3) = 2 pounds.
Rubric:
[2] 2 and appropriate work is shown.


6.
6. A high school drama club is putting on their annual theater production. There is a maximum of
   800 tickets for the show. The costs of the tickets are $6 before the day of the show and $9 on
   the day of the show. To meet the expenses of the show, the club must sell at least $5,000 worth
   of tickets.
a)   Write a system of inequalities that represent this situation.
b)   The club sells 440 tickets before the day of the show. Is it possible to sell enough
     additional tickets on the day of the show to at least meet the expenses of the show? Justify
     your answer.
Answer: MODEL ANSWER GIVEN BELOW
Key: a) x + y ≤ 800
        6x + 9y ≥ 5000
    b) Yes with appropriate work shown to justify the answer.
Commentary: This question aligns to A.CED.3 because a student writes a system of
            inequalities to determine a viable solution.
Mathematical Practices: 4 and 6
Rationale:
     a) Let x = number of presale tickets
            y = number of day of show tickets
       x + y ≤ 800
       6x + 9y ≥ 5000
     b) 6(440) + 9y ≥ 5000
         2640  + 9y ≥ 5000
                 9y ≥ 2360
                  y ≥ 262.2
          263 tickets
          440 advance purchase tickets added to 263 day of show tickets is 703 tickets, which
          is below the 800 ticket maximum. So yes, it is possible.
Rubrics:
(a) [2]    x + y ≤ 800 and 6x + 9y ≥ 5000 .
(b) [2]   Yes, and appropriate work is shown.


7.
7. During a snowstorm, a meteorologist tracks the amount of accumulating snow. For the first
   three hours of the storm, the snow fell at a constant rate of one inch per hour. The storm then
   stopped for two hours and then started again at a constant rate of one-half inch per hour for the
   next four hours.
    a)   On the grid below, draw and label a graph that models the accumulation of snow over time
         using the data the meteorologist collected.
b)   If the snowstorm started at 6 p.m., how much snow had accumulated by midnight?
Answer: MODEL ANSWER GIVEN BELOW
Key: a) See graph in rationale below.
     b) 3 ½
Commentary: This question aligns to F.IF.4 because the students sketch a graph based on a
            verbal description of the snowstorm.
Mathematical practices: 4
Rationale:
Rubric:
[4] A correct graph is drawn, the axes are labeled correctly, and 3 ½ is stated.


8.
8. Next weekend Marnie wants to attend either carnival A or carnival B. Carnival A charges $6 for
   admission and an additional $1.50 per ride. Carnival B charges $2.50 for admission and an
   additional $2 per ride.
   a)   In function notation, write A(x) to represent the total cost of attending carnival A and
        going on x rides. In function notation, write B(x) to represent the total cost of attending
        carnival B and going on x rides.
b)   Determine the number of rides Marnie can go on such that the total cost of attending each
     carnival is the same. [Use of the set of axes below is optional.]
c)   Marnie wants to go on five rides. Determine which carnival would have the lower total
     cost. Justify your answer.

Answer: MODEL ANSWER GIVEN BELOW
Key: a) A(x) = 1.50x + 6
        B(x) = 2x + 2.50
     b) 7 rides
     c) Carnival B with appropriate justification.
Commentary: This question aligns to A.REI.11 because the answer to the problem requires
            the student to solve A(x) = B(x), either algebraically or graphically.
Rationale:
    a)     A(x) = 1.50x + 6
           B(x) = 2x + 2.50
    b)     A(x) = B(x)
           1.50x + 6 = 2x + 2.5
               x=7
     c)    Carnival A cost = 1.50x + 6
                           = 1.50(5) + 6
                           = $13.50
           Carnival B cost = 2x + 2.50
                           = 2(5) + 2.50
                           = $12.50
           Carnival B because it costs $12.50 and carnival A costs $13.50.
Rubrics:
(a) [2]    A(x) = 1.50x + 6 and B(x) = 2x + 2.50
(b) [2]   7 and appropriate work is shown.
(c) [2] Carnival B and an appropriate justification is given, such as showing that carnival B
        costs $12.50 and carnival A costs $13.50.



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