Algebra Problems of the Day (Integrated Algebra Regents, January 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Integrated Algebra Regents, January 2012

Part I: Each correct answer will receive 2 credits.


1. Which expression is equivalent to 64 - x²?

1) (8 - x)(8 - x)
2) (8 - x)(8 + x)
3) (x - 8)(x - 8)
4) (x - 8)(x + 8)

Answer: 2) (8 - x)(8 + x)

The Difference of Two Perfect Squares factors into their conjugates.

Choice (1) is 64 - 16x + x²

Choice (3) is x² - 16x + 64

Choice (4) is x² - 64, which is the inverse of what we want.

2. Mr. Smith invested $2,500 in a savings account that earns 3% interest compounded annually. He made no additional deposits or withdrawals. Which expression can be used to determine the number of dollars in this account at the end of 4 years?

1) 2500(1 + 0.03)⁴
2) 2500(1 + 0.3)⁴
3) 2500(1 + 0.04)³
4) 2500(1 + 0.04)³

Answer: 1) 2500(1 + 0.03)⁴

The formula is A = P(1 + r)^t, where r is the rate as a decimal and t is the time in years. In this example, 3% means r = 0.03 and t = 4 years.

Choice (2) has 30% interest for 4 years, which is great if you can find it.

Choice (3) has 4% interest for 3 years.

Choice (4) has 40% interest for 3 years.

3. What is 2√(45) expressed in simplest radical form?

1) 3√(5)
2) 5√(5)
3) 6√(5)
4) 18√(5)

Answer: 3) 6√(5)

Factor 45 into 9 * 5. Since 9 is a perfect square, √(9) = 3, and 2 * 3 = 6. This leaves only the factor of 5 under the radical sign.

4. Which graph does not represent a function?

Answer: 3) (see image)

Choice (3) fails the Vertical Line Test. There are values of x that have two distinct y values for outputs. You cannot have that in a function.

For example, the points (1, 1) and (1, -1) are both on the graph in Choice (3).

Choice (4) may be a piecewise graph, but it is still a function.

5. Timmy bought a skateboard and two helmets for a total of d dollars. If each helmet cost h dollars, the cost of the skateboard could be represented by

1) 2dh
2) dh/2
3) d - 2h
4) d - h/2

Answer: 3) d - 2h

He spent d dollars for eaverything. He spent 2h for the two helmets (they were h each). Whatever is left over must be the cost of the skateboard. So subtract d - 2h to find the skateboard.

More to come. Comments and questions welcome.

More Regents problems.

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