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.


11. A student correctly graphed the parabola shown below to solve a given quadratic equation.

What are the roots of the quadratic equation associated with this graph?

1) -6 and 3
2) -6 and 0
3) -3 and 2
4) -2 and 3

Answer: 4) -2 and 3

The roots are the zeroes of the function. For what values of x is the function (the y value) equal to 0?

The x-intercepts occur at -2 and 3. This is Choice (4).

The y-intercept is -6, which in this graph is NOT the minimum, so it isn't really an important number on the graph, compared to the roots and the vertex.(in my opinion)

12. Which value of x is the solution of the equation 2/3 x + 1/2 = 5/6

1) 1/2
2) 2
3) 2/3
4) 3/2

Answer: 1) 1/2

Plugging in choices may be messy if you don't like dealing with fractions. It may be easier to solve the equation.

(2/3 x + 1/2 = 5/6 ) * 6

(6)(2/3)x + (6)(1/2) = (6)(5/6)

4x + 3 = 5

4x = 2

x = 1/2

The correct answer is Choice (1).

13. What is the range of the data represented in the box-and-whisker plot shown below?

1) 40
2) 45
3) 60
4) 100

Answer: 3) 60

The range is the highest number minus the lowest number. The highest number is 75 and the lowest is 15. And 75 - 15 = 60, which is Choice (3).

Choice (1), 40, is the median, the middle of the data. If is also the length of the box portion of the graph, also known as the Interquartile Range.

Choice (2), 45, is nothing of significance. It's the middle of the number line, but for purposes of statistics, that is irrelevant.

Choic (4), 100, is the range shown in the graph, but the data doesn't cover the entire number line. The line could have stopped at 90 or 80 and nothing on the graph would have changed.

14. Which equation illustrates the associative property?

1) x + y + z = x + y + z
2) x(y + z) = xy + xz
3) x + y + z = z + y + x
4) (x + y) + z = x + (y + z)

Answer: 4) (x + y) + z = x + (y + z)

The Associative Property tells us that we can group the terms differently, like in Choice (4).

Choice (1) shows the Identity Property. The left side is the same as the right side of the equal sign.

Choice (2) shows the Distributive Property of Multiplication Over Addition.

Choice (3) shows the Commutative Property, because the order of the operations has been switched around.

15. Josh and Mae work at a concession stand. They each earn $8 per hour. Josh worked three hours more than Mae. If Josh and Mae earned a total of $120, how many hours did Josh work?

1) 6
2) 9
3) 12
4) 15

Answer: 2) 9

Use the equation 8(x + x + 3) = 120 to find x, the number of hours that MAE worked, and then x + 3, which is how many JOSH worked. (You could also use 8(x + x - 3) = 120 to find Josh immediately, if you realize that you can use subtraction here.)

What is inside the parentheses are the total number of hours worked. The 8 is their pay since they are both paid the same amount. You don't need to distribute it -- you can just divide the equation by 8!

8(x + x + 3) = 120

2x + 3 = 15

2x = 12

x = 6

MAE worked 6 hours, so JOSH work 6 + 3 = 9 hours.

If you didn't and the 3, you got Choice (1). If you just divided by 8, you got Choice (4). If you divided by 8 and then subtracted 3, you got Choice (3). Those would all be bad.

More to come. Comments and questions welcome.

More Regents problems.

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