Wednesday, December 01, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 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, August 2011

11. What are the vertex and the axis of symmetry of the parabola shown in the graph below?

1) vertex: (1,6); axis of symmetry: y = 1
2) vertex: (1,6); axis of symmetry: x = 1
3) vertex: (6,1); axis of symmetry: y = 1
4) vertex: (6,1); axis of symmetry: x = 1

Answer: 2) vertex: (1,6); axis of symmetry: x = 1

The axis of symmetry is the imaginary vertical line that divides the parabola in half, intersecting the vertex. All vertical lines have an equation that starts "x = ", so eliminate Choices (1) and (3).

Without even looking at the graph, you can see that Choice (4) is incorrect because it says x = 1 but also has an x-coordinate of 6, which makes no sense.

The vertex is at (1,6), so the correct answer is Choice (2).

Choice (2) is the only statement that is false.

12. The diagram below shows right triangle ABC.

Which ratio represents the tangent of ∠ABC?

1) 5/13
2) 5/12
3) 12/13
4) 12/5

The tangent ratio is defined as the length of the leg opposite from the angle over the length of the leg adjacent to the angle. That makes is 5/12, which is Choice (2).

Tangent is the one ratio that doesn't use the hypotenuse and can have a value greater than 1.

13. What is the value of the expression −3x2y + 4x when x = −4 and y = 2?

1) -112
2) -80
3) 80
4) 272

Substitute and evaluate using the Order of Operations. You can do the entire thing in a calcuator as long as you enter it correctly.

-3(-4)2(2) + 4(-4) = -112

The incorrect choices will happen if you evaluate this by hand and do something out of sequence such as multiply by -3 or 3 before squaring.

14. Which expression is equivalent to −3x(x − 4) − 2x(x + 3)?

1) −x2 − 1
2) −x2 − 18x
3) −5x2 − 6x
4) −5x2 + 6x

Use the Distributive Property and Combine Like Terms. Be careful with distributing the negative sign.

−3x(x − 4) − 2x(x + 3)

−3x2 + 12x − 2x2 - 6x

−5x2 + 6x

15. The data in the table below are graphed, and the slope is examined.

The rate of change represented in this table can be described as

1) negative
2) positive
3) undefined
4) zero

As the x values increase, the y values are decreasing. Therefore the rate of change is negative.

For the rate of change to be positive, both x and y would have to increase together or decrease together. (If they decrease together, that means that they are increasing together when you read the table from the other direction.)

A zero rate of change means that the y values are not changing at all, regardless of the x values. This would result in a horizontal line.

An undefined rate of change would be if the x-value remained constant by the y-values increased or decreased. This would result in a vertical line and would not be a function.

More to come. Comments and questions welcome.

More Regents problems.

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