Algebra Problems of the Day (Integrated Algebra Regents, January 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, January 2011

Part I: Each correct answer will receive 2 credits.


1. Given:

X = {1, 2, 3, 4}
Y = {2, 3, 4, 5}
Z = {3, 4, 5, 6}

What is the intersection of sets X, Y, and Z ?

1) {3, 4}
2) {2, 3, 4}
3) {3, 4, 5}
4) {1, 2, 3, 4, 5, 6}

Answer: 1) {3, 4}

The intersection of the three sets are the elements that can be found within all three sets.

Only 3 and 4 are in all three of the sets. This is Choice (1).

That is choice (3).

Choice (2) is the intersection of X and Y.

Choice (3) is the intersection of Y and Z.

Choices (4) is the union of X, Y, and Z, and contains all the elements that could be found in ANY of the three sets.

2. Which graph could be used to find the solution of the system of equations y = 2x + 6 and y = x² + 4x + 3?

Answer: 4) [See image]

The parabola opens upward (and has a y-intercept of 3) so eliminate Choices (2) and (3). The straight line has a slope of 2 and y-intercept of 6, so eliminate Choice (1).

Only Choice (4) remains.

3. What is the relationship between the independent and dependent variables in the scatter plot shown below?

1) undefined correlation
2) neagtive correlation
3) positive correlation
4) no correlation

Answer: 3) positive correlation

As x gets higher, y generally goes higher. That is a positive correlation.

There is no such thing as an undefined correlation.

4. Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate?

1) 2 + (x + 4)
2) 2x + 4
3) 2(x + 4)
4) 4(x + 2)

Answer: 3) 2(x + 4)

T = A + 4 and B = 2T

So B = 2(A + 4). Using x instead of A, you have Choice (3).

5. 3) 2(x + 4)

1) {(3/4,0), (0,1), (3/4 ,2)}
2) {(-2,2), (-1/2,1), (-2,4)}
3) {(-1,4), (-0,5), (0,4)}
4) {(2,1), (4,3), (6,5)}

Answer: 4) {(2,1), (4,3), (6,5)}

In a function, every input can only have one output. The input cannot repeat with a different output number. (And since this is written as a set, it shouldn't repeat with the same output number either -- but that's set notation, not the definition of a function.)

The output can repeat with different inputs.

In Choice (1), x = 3/4 repeats. In Choice (2), x = -2 repeats. In Choice (3), x = 0 repeats.

Choice (4) has no repeats in the input.

More to come. Comments and questions welcome.

More Regents problems.

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