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.


6. 6 The graph of y = |x + 2| is shown below.

Which graph represents y = -|x + 2|?

Answer: 4) [See Image]

The negative coefficient in the front will cause the absolute value graph to open downward from its center at (0,-2) instead of upward., which is the inverse of what we want.

7. Two equations were graphed on the set of axes below.

Which point is a solution of the system of equations shown on the graph?

1) (8,9)
2) (5,0)
3) (0,3)
4) (2,-3)

Answer: 1) (8,9)

The solutions are the points where the two lines intersect. One of those points is at (8,9) and the other solution is at non-Integer coordinates that can not be determined easily from a graph.

Choice (2) is one of the roots of the parabola, not a solution to the system.

Choice (3) is the y-intercept for the linear graph, not a solution to the system.

Choice (4) is the vertex (minimum) of the parabola, not a solution to the system.

8. Byron is 3 years older than Doug. The product of their ages is 40. How old is Doug?

1) 10
2) 8
3) 5
4) 4

Answer: 3) 5

It's probably quickest to multiply from the choices. Remember that Byron is older than Doug, and we are looking for the younger of the two.

Choice (1) 10 * 13 = 130 =/= 40

Choice (2) 8 * 11 = 88 =/= 40

Choice (3) 5 * 8 = 40

Choice (4) 4 * 7 = 28 =/= 40

The algebraic method, when there aren't choices, is as follows:

x(x + 3) = 40
x² + 3x = 40
x² + 3x - 40 = 0
(x + 8)(x - 5) = 0
x + 8 = 0 or x - 5 = 0
x = -8 or x = 5
Discard the negative age, Doug is 5.

9. The actual dimensions of a rectangle are 2.6 cm by 6.9 cm. Andy measures the sides as 2.5 cm by 6.8 cm. In calculating the area, what is the relative error, to the nearest thousandth?

1) 0.055
2) 0.052
3) 0.022
4) 0.021

Answer: 2) 0.052

To find the relative error, find the actual error -- the difference between the actual and calculated results -- and divide it by the actual measurement.

The area Andy calculated is 2.5 * 6.8 = 17.

The actual area is 2.6 * 6.9 = 17.94.

The difference between the two measurements is 0.94, which when divided by the actual area gives a relative error of 0.94/17.94 = 0.0523...

10. Which graph represents the inequality y > 3?

Answer: 1) {See Image]

If we want "greater then" but NOT "greater than or equal to", we want a graph with a solid line.

We want y values, which means we care about the numbers going up the y-axis, not the numbers going to the right on the x-axis.

We want Choice (1).

More to come. Comments and questions welcome.

More Regents problems.

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