Tuesday, February 06, 2018

January 2018 Common Core Algebra I Regents, Part 1 (mult choice)

The following are some of the multiple questions from the recent January 2018 New York State Common Core Algebra I Regents exam.
The answers to Part II can be found here
The answers to Parts III and IV can be found here

January 2018 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.


1. When solving the equation 12x2 - 7x = 6 - 2(x2 - 1), Evan wrote computations. 12x2 - 1x = 6 - 2x2 + 2 as his first step. Which property justifies this step?

Answer: (4) distributive property of multiplication over subtraction.


2. Jill invests $400 in a savings bond. The value of the bond, V(x), in hundreds of dollars after x years is illustrated in the table below.

x V(x)
0 4
1 5.4
2 7.29
3 9.84

Which equation and statement illustrate the approximate value of the bond in hundreds of dollars over time in years?

Answer: (3) V(x) = 4(1.35)x and it grows.
Choices (1) and (2) are no good because the base is less than 1.00, which would cause decay, not growth. Choice (4) is out because it is not decay.


3. Alicia purchased H half-gallons of ice cream for $3.50 each and P packages of ice cream cones for $2.50 each. She purchased 14 items and spent $43. Which system of equations could be used to determine how many of each item Alicia purchased?

Answer:(1) 3.50H + 2.50P = 43, H+P=14
The sum of the prices is $43.00. The total number of items is 14.


4. A relation is graphed on the set of axes below.


Based on this graph, the relation is

Answer: (2) a function because it passes the vertical line test


5. Ian is saving up to buy a new baseball glove. Every month he puts $10 into a jar. Which type of function best models the total amount of money in the jar after a given number of months?

Answer: (1) linear
There is a constant rate of change in the amount he has saved.


6. Which ordered pair would not be a solution to y = x3 - x?

Answer: (4) (-1,-2)
(-1)3 - (-1) = 0, not -2.


7. Last weekend, Emma sold lemonade at a yard sale. The function P(c) = .50c - 9.96 represented the profit, P(c), Emma earned selling c cups of lemonade. Sales were strong, so she raised the price for this weekend by 25 cents per cup. Which function represents her profit for this weekend?

Answer: (4) P(c) = .75c - 9.96
Raising the price per cup would mean increasing the cost per cup from 50 cents (or .50) to 75 cents (or .75).


8. The product of Sqrt(576) and Sqrt(684) is

Answer: (3) irrational because one factor is irrational
Sqrt(576) is 24, which is rational. Sqrt(684) is irrational. Their product is irrational.


9. Which expression is equivalent to y4 - 100?

Answer: (3) (y2 + 10)(y2 - 10)
Difference of Squares rule.


10. The graphs of y = x2 - 3 and y = 3x - 4 intersect at approximately

Answer: (3) (0.38, -2.85) and (2.62, 3.85)
If you graph both functions, you will see that they have two points of intersection, so choices (1) and (2) are out. The second one occurs when x is less than 3, so choice (4) is out.
You can also put the x values into each equation and check the y values.


11. The expression -4.9t2 + 50t + 2 represents the height, in meters, of a toy rocket t seconds after launch. The initial height of the rocket, in meters, is

Answer: (2) 2
When t = 0, the expression is 0 + 0 + 2 = 2.
Note: The -4.9 will show up in gravity questions when meters are the units being used. Sometimes it's rounded to -5.


12. If the domain of the function f(x) = 2x2 - 8 is {-2, 3, 5}, then the range is

Answer: (3) {O, 10, 42}
Simple Order of Operations question.
Calculator tip: type 2x^{-2,3,5} - 8 into your graphing calculator and it will give you the range in one shot.


13. Which polynomial is twice the sum of 4x2 - x + 1 and -6x2 + x - 4?

Answer: (3) -4x2 - 6
Twice the sum means add them first then double it.
(4x2 - x + 1) + (-6x2 + x - 4) = -2x2 - 3, (the middle term cancelled out)
2 (-2x2 - 3) = -4x2 - 6


14. What are the solutions to the equation 3(x - 4)2 = 27?

Answer: (1) 1 and 7

3(x - 4)2 = 27
(x - 4)2 = 9
x - 4 = 3 or x - 4 = -3
x = 7 or x = 1



15. A system of equations is shown below.

Equation A: 5x + 9y = 12
Equation B: 4x - 3y = 8

Which method eliminates one of the variables?

Answer: (2) Multiply equation B by 3 and add the result to equation A.
9y + (-9y) will eliminate the y variable so you can solve for x.


16. The 15 members of the French Club sold candy bars to help fund their trip to Quebec. The table below shows the number of candy bars each member sold.


When referring to the data, which statement is false?

Answer: (1) The mode is the best measure of central tendency for the data.
The median (middle value when data is sorted) is 53. The range is 120 - 0 = 120. There are two outliers: 0 and 120, which are very low and very high compared to the rest of the data. The mode is 68, which comes from 3 of the 5 highest values of the 15 pieces of data. That is not the best measure of central tendency to use for this data.


17. Given the set {x| -2 < x < 2, where x is an integer}, what is the solution of -2(x - 5) < 10?

Answer: (2) 1, 2
You could check 0, 1, -1 and have enough information to find the correct choice. (You wouldn't need -1, in fact, in this case.) Or you can simplify the inequality first.

-2(x - 5) < 10
x - 5 > -5
x > 0



18. If the pattern below continues, which equation(s) is a recursive formula that represents the number of squares in this sequence?

Answer: (3) a1 = 3, an = an-1 + 2
Choices (1) and (2) are not recursive. The first design, a1 has 3 squares in it, so choice (4) is incorrect.


19. If the original function f(x) = 2x2 - 1 is shifted to the left 3 units to make the function g(x), which expression would represent g(x)?

Answer: (2) 2(x + 3)2 - 1
Shifting the vertex 3 units left gives you "(x + 3)". A shift the the right would be "(x - 3)".
Choices (3) and (4) represent shifts up and down, respecitvely.


20. First consider the system of equations y = (-1/2)x+ 1 and y = x- 5.
Then consider the system of inequalities y > (-1/2)x + 1 and y < x- 5.
When comparing the number of solutions in each of these systems, which statement is true?

Answer: (3) The system of inequalities has more solutions.
The system of equations has a single solutions. Two linear equations meet at a single point unless they have the same slope. The first line has a slope of -1/2, the second 1.
This system of inequalities has an infinite number of solutions; e.g., (6,0), (7,0), (8,0), etc.


21. Nora inherited a savings account that was started by her grandmother 25 years ago. This scenario is modeled by the function A(t) = 5000(1.013)t + 25, where A(t) represents the value of the account, in dollars, t years after the inheritance. Which function below is equivalent to A(t)?

Answer: (4) A(t) = 5000(1.013)t (1.013)25
When you multiply two expressions that have the same base, you add the exponents. The reverse is true: if you have a base with an exponent, you can factor it by keeping the base and reducing the exponents to two addends (i.e., to two numbers or expressions that add up to the original expression).
So, for example, x8 = x5 * x3, or
(m + 3)n+5 = (m + 3)n * (m + 3)5


22. The value of x which makes

(2/3)((1/4)x - 2) = (1/5)((4/3)x - 1)
true is

Answer: (4) -11.3... (repeating decimal; i.e., 1/3)

Multiple both side by 3*5, which is 15
(15)(2/3)((1/4)x - 2) = (15)(1/5)((4/3)x - 1)
(10)((1/4)x - 2) = (3)((4/3)x - 1)
2.5x - 20 = 4x - 3
-17 = 1.5x
-11.3... = x



23. Which quadratic function has the largest maximum over the set of real numbers?

Answer: (2) k(x)


Choice (3) is in vertex form. It's maximum point occurs at (5, 5).
Choice (1) can be entered into the graphing calculator, or you can use x = -b/a = -2/-1 = 1
f(1) = -(1)2 + 2(1) + 4 = -1 + 2 + 4 = 5, which gives (1, 5). So neither (1) or (3) can be the answer.
Choice (4) does not show a value higher than 3, but neither of those two points are the vertex. You could do a quadratic regression to find the function and then find the vertex, but look at the rate of change in the table. It goes +6, +4, +2, ... getting smaller with each step. You can deduce that h(1.5) will NOT be greater than 5.
Choice (2), like (4) doesn't show the vertex. However, the highest number it does show is 5. The vertex must be higher than 5. That makes choice (2) the answer.


24. Voting rates in presidential elections from 1996-2012 are modeled below.


Which statement does not correctly interpret voting rates by age based on the given graph?

Answer: (2) From 1996-2012, the average rate of change was positive for only two age groups.
The rate of change was negative for the "45 to 64 years" age group, decreasing from 68.2 to 67.9. It increased for the other three groups from 1996-2012.

End of Part I

How did you do?

Questions, comments and corrections welcome.

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