Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2014)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2014

Part I: Each correct answer will receive 2 credits.


21. The expression (CBRT(27²)(CBRT(16⁴) is equivalent to

1) 12x² (CBRT(2))
2) 12x (CBRT(2x))
3) 6x (CBRT(2x²))
4) 6x² (CBRT(2))

Answer: 4) 6x² (CBRT(2))

Combine the cube roots by multiplying and then factor the result, looking for groups of three.

27 x² * 16 x⁴

27 * 16 * x⁶

3 * 3 * 3 * 2 * 2 * 2 * 2 * x² * x² * x²

Remove three factors of 3, three factors of 2, three factors of x², and you are left with just 2 under the radical sign.

Outside of the radical you will have 3 * 2 * x², or 6x².

22. 2 Which sketch shows the inverse of y = a^x, where a > 1?

Answer: 3) (see image)

The inverse to y = a_x would be the graph of x = a^y, or y = log_ax.

Choice (1) shows y = a^x

Choice (2) shows y = a^x where a < 1

CHoice (4) shows y = a² after it's been transposed over y = x.

23. The expression (x² + 9x - 22) / (x² - 121) divided by (2 - x) is equivalent to

1) x - 11
2) 1 / (x - 11)
3) 11 - x
4) 1 / (11 - x)

Answer: 4) 1 / (11 - x)

Factor everything and eliminate identical factors from the numerator and the denominator.

(x² + 9x - 22) / (x² - 121) * 1 / (2 - x)
= ( (x + 11)(x - 2) ) / ( (x - 11)(x + 11) * 1 / ( (-1)(x - 2) )
= ( (x + 11)(x - 2) ) / ( (x - 11)(x + 11) * 1 / ( (-1)(x - 2) )
= ( 1 ) / (x - 11) * 1 / ( (-1) )
= 1 / ( (-1)(x - 11) )
= 1 / (11 - x)

24. Which graph represents the solution set of (x + 16) / (x - 2) < 7?

Answer: 3) see image

If you graph y = (x + 16) / (x - 2) and look where it will be less or equal to 7, you will see that there is an asymptote at x = 2. As x approaches 2 from the right side, y goes to infinity, and as x approaches 2 from the left side, y goes to negative infinity.

The entire left side of the graph, when x has a value less than but NOT equal to negative two, has y-coordinates less than seven. On the right side, y will be less than or equal to 7 when x is greater than or equal to five.

Algebraically, it takes a little longer. We need to have a single fraction on the left side and a 0 on the right side.

(x + 16) / (x - 2) < 7
(x + 16) / (x - 2) - 7 < 0
(x + 16) / (x - 2) - 7(x - 2)/(x - 2) < 0
(x + 16) / (x - 2) - (7x - 14)/(x - 2) < 0
(-6x + 30) / (x - 2) < 0

Find the zeroes of the numerator and the denominator

-6x + 30 = 0
-6x = -30
x = 5

x - 2 = 0
x = 2

Check the intervals (-&inf;, 2), (2,5), and (5, &inf;).
I'll pick 0, 3 and 10 to check.
(0 + 16) / (0 - 2) = 16 / -2 = -8, which is less than 7. This is true.
(3 + 16) / (3 - 2) = 19 / 1, which is NOT less than 7. This interval is NOT part of the solution.
(10 + 16) / (10 - 2) = 26 / 8 = 3.25, which is less than 7.

Also, x =/= 2 because this is undefined, but x = 5 is a solution.

The intervals are x < 2 and x > 5.

More to come. Comments and questions welcome.

More Regents problems.

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