## Friday, October 22, 2021

### Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)

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, June 2012

16. Which expression is equivalent to (n°m°p)(x), given m(x) = sin x, n(x) = 3x, and p(x) = x2?

1) sin(3x)2
2) 3 sin x2
3) sin2(3x)
4) 3 sin2x

As with compositions of transformations, when you have a composition of functions, you work from right to left. You want to find n OF m OF p OF x, or more explicitly, n(m(p(x))).

p(x) = x2

m(x2) = sin x2

n(sin x2) = 3 sin x2

The other choices will occur if you do operations in the wrong order.

17. The value of csc 138°23' rounded to four decimal places is

1) -1.3376
2) -1.3408
3) 1.5012
4) 1.5057

If you can't calculate that directly in your calculator, remember that CSC is 1/SIN.

Since Sin is positive between 0 and 180°, you can eliminate Choices (1) and (2). (Without checking, my guess is that both of these are related to Cosine.)

23/60 = .383333333

1/SIN(138.38333333) = 1/0.664143715 = 1.50569821 = 1.5057

18. Which function is one-to-one?

1) k(x) = x2 + 2
2) g(x) = x3 + 2
3) f(x) = |x| + 2
4) j(x) = x4 + 2

Answer: 2) g(x) = x3 + 2

For a relation to be a function, every x can have only 1 value associated with it. It must pass the Vertical Line Test.

For a function to be one-to-one, every y value must be associate with exactly 1 x value. It must pass a horizontal line test as well.

Choices (1), (3) and (4) are all U or V shaped. They come down from infinity, hit a minimum point and turn back up toward infinity. Every y value, except the minimum, will be associated with both a positive AND a negative x value.

19. The conjugate of the complex expression -5x + 4i is

1) 5x - 4i
2) 5x + 4i
3) -5x - 4i
4) -5x + 4i

When it came to binomials, and the Difference of Squares rule, (x + y)(x - y) were conjugates. The two terms were the same but the sign in the middle was change from + to -.

With complex numbers, it is the same thing: (-5x + 4i) and (-5x - 4i) are conjugates. The sign in the middle -- the sign before the imaginary part of the complex number -- changes. The sign of the Real portion does not change.

20. What is a positive value of tan 1/2 x, when sin x = 0.8?

1) 0.5
2) 0.4
3) 0.33
4) 0.25

The value of of sin x = 0.8 when x = 51.130... degrees, as in a 6-8-10 right triangle.

Half of 51.130... is 26.565..

Tan 25.565 = 0.478..., which rounds to .5.

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.