Tuesday, September 07, 2021

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 = ax, where a > 1?


Answer: 3) (see image)


The inverse to y = ax would be the graph of x = ay, or y = logax.

Choice (1) shows y = ax

Choice (2) shows y = ax 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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