Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

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, August 2010

Part I: Each correct answer will receive 2 credits.


11. The width of a rectangle is 3 less than twice the length, x. If the area of the rectangle is 43 square feet, which equation can be used to find the length, in feet?

1) 2x(x - 3) = 43
2) x(3 - 2x) = 43
3) 2x + 2(2x - 3) = 43
4) x(2x - 3) = 43

Answer: 4) x(2x - 3) = 43

The area of a rectangle is the length times the width. The width is 2L - 3.

So A = LW = L(2L - 3) = 43.

Substitute x for L and you have Choice (4).

12. Which value of x is the solution of (2x - 3) / (x - 4) = 2/3?

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

Answer: 2) 1/4

You can cross-multiply and solve, or you can substitute each of the choices.

(2x - 3) / (x - 4) = 2/3

3(2x - 3) = 2(x - 4)

6x - 9 = 2x - 8

4x = 1

x = 1/4

Check: (2(1/4) - 3) = -2.5, 1/4 - 4 = -3.75.
-2.5/-3.75 = 2/3. CHECK!

13. What is the perimeter of a regular pentagon with a side whose length is x + 4?
1) x² + 16
2) 4x + 16
3) 5x + 4
4) 5x + 20

Answer: 4) 5x + 20

A regular pentagon has 5 sides of equal length. If one side is x + 4, then the perimeter is 5(x + 4), or 5x + 20.

14. Which equation represents a line parallel to the y-axis?

1) x = y
2) x = 4
3) y = 4
4) y = x + 4

Answer: 2) x = 4

The y-axis has an equation of x = 0. It is parallel to all lines with equations x = a, where a is a real number.

Choice (1) x = y will intersect the y-axis when y = 0.

Choice (3) y = 4 is a horizontal line, not a vertical line. It will intersect the y-axis at (0,4).

Choice (4) y = x + 4 is not a vertical line. It will intersect the y-axis when y = 4.

15. The diagram below shows the graph of y = -x² - c.

Which diagram shows the graph pf y = x² - c?

Answer: 1) [SEE IMAGE]

A quadratic equation starting with y = -x² will create a parabola opening downward. A quadratic equation starting with y = x² will create a parabola opening upward. Eliminate Choices (2) and (3).

The y-intercept hasn't changed between the two equations. They are both at -c, where c appears to be a postivie number. Only Choice (1) has shares the same y-intercept with the given graph, so it must be the answer.

More to come. Comments and questions welcome.

More Regents problems.

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