August 2023 Algebra 2 Regents, Part III

This exam was adminstered in August 2023.

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Algebra 2 August 2023

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. 3 Sketch p(x) = -log₂(x + 3) + 2 on the axes below.
Describe the end behavior of p(x) as x → -3.
Describe the end behavior of p(x) as x → ∞.

Answer:

Put the equation into your graphing calculator and look at the table of values. If you release that log₂(1) is equal to 0, then you can deduce that p(-2) = 0 + 2, so (-2,2) is a point on the graph.

If you plot the points: (-2,2), (-1,1), (0,0.415), (1,0), (2,-0.322), and (3,-0.585), you'll have enough for a good sketch. There will be a vertical asymptote at x = -3.

Your graph should look like the following:

As x goes to -3, p(x) goes to infinity. As x goes to infinity, p(x) goes to negative infinity.

34. Solve for x algebraically: 1 / (x - 6) + x / (x - 2) = 4 / (x² - 8x + 12)

Answer:

You need to combine the fractions on the left if you wish to cross-multiply. Or you can multiply both sides of the equation by (x² - 8x + 12) if you realize that the polynomial factors into (x - 6)(x - 2).

1 / (x - 6) + x / (x - 2) = 4 / (x² - 8x + 12)
(x - 6)(x - 2) ( 1 / (x - 6) + x / (x - 2) ) = (4 / (x² - 8x + 12)) (x - 6)(x - 2)
x - 2 + x(x - 6) = 4
x - 2 + x² - 6x = 4
x² - 5x - 6 = 0
(x - 6)(x + 1) = 0
x - 6 = 0 or x + 1 = 0
x = 6 or x = -1

Throw out x = 6 as extraneous because 1 / (6 - 6) is undefined.

The only solution is x = -1.

35. Solve the following system of equations algebraically for x, y, and z.
2x + 4y - 3z = 12
3x - 2y + 2z = -9
-x + y - 3z = 0

Answer:

You can rewrite the third equation to solve for either x or y and then substitute the expression into the first two equations. Then you can solve that system.

-x + y - 3z = 0
y - 3z = x

2(y - 3z) + 4y - 3z = 12
3(y - 3z) - 2y + 2z = -9

2y - 6z + 4y - 3z = 12
3y - 9z - 2y + 2z = -9

6y - 9z = 12
y - 7z = -9

6y - 9z = 12
y = 7z - 9

6(7z - 9) - 9z = 12
42z - 54 - 9z = 12
33z = 66
z = 2

y = 7(2) - 9 = 5

x = y - 3z = (5) - 3(2) = -1

36. Two classes of students were entered into an experiment to see whether using an interactive whiteboard leads to better grades. It was observed that the mean grade of students in the class with the interactive whiteboard was 0.6 points higher than the class without it. To determine if the observed difference is statistically significant, the classes were rerandomized 5000 times to study these random differences in the mean grades. The output of the simulation is summarized in the histogram below.

Determine an interval containing the middle 95% of the simulation results. Round your answer to the nearest hundredth.
Does the interval indicate that the difference between the classes’ grades is significant? Explain.

Answer:

The interval is the mean plus or minus twice the standard deviation.

0.01 + 2(0.38) = 0.77
0.01 - 2(0.38) = -0.75

The interval is [-0.75, 0.77]

Since 0.6 is within the interval, so it's not significant.

End of Part III

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