Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

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 2013

Part I: Each correct answer will receive 2 credits.


1. What is the equation of the graph shown below?

1) y = 2^x
2) y = 2^-x
3) x = 2^y
4) x = 2^-y

Answer: 2) y = 2^-x

The graph shows exponential decay, which is given by the formula y = a^x, where 0 < a < 1.

The base in this graph cannot be 2, but it could be 1/2, which is 2^-1.

Checking the graph you can see that 2^{-(-1) = 2, 2-(-2) = 4, and 2-(-3) = 8.}

2. Which ordered pair is a solution of the system of equations shown below?

x + y = 5
(x + 3)² + (y - 3)² = 53

1) (2,3)
2) (5,0)
3) (-5,10)
4) (-4,9)

Answer: 3) (-5,10)

A quick check will tell you that all four points are solutions for x + y = 5, so we can ignore that one and focus on the other equation.

(2 + 3)² + (3 - 3)² = 5² + 0² = 25

(5 + 3)² + (0 - 3)² = 8² + (-3)² = 73

(-5 + 3)² + (10 - 3)² = (-2)² + 7² = 53

(-4 + 3)² + (9 - 3)² = (-1)² + 6² = 37

3. The relationship between t, a student’s test scores, and d, the student’s success in college, is modeled by the equation d = 0.48t + 75.2. Based on this linear regression model, the correlation coefficient could be

1) between -1 and 0
2) between 0 and 1
3) equal to -1
4) equal to 0

Answer: 2) between 0 and 1

It is a positive correlation, so the correlation coefficient must be a positive number between 0 and 1.

It wouldn't be negative, so choices (1) and (3) are eliminated. A coefficient of 0 means that there is no correlation at all, so there would be no equation that could model it.

4. What is the common ratio of the geometric sequence shown below?

-2, 4, -8, 16, …

1) -1/2
2) 2
3) -2
4) -6

Answer: 3) -2

To find the common ratio divide any term by the term before it.

4/(-2) = -8/2 = 16/(-8) = -2

S_n = a₁(1 - rⁿ) / (1 - r)
= 3(1 - (-4)⁸) / (1 - (-3))
= -39321

5. Given the relation {(8,2), (3,6), (7,5), (k,4)}, which value of k will result in the relation not being a function?

1) 1
2) 2
3) 3
4) 4

Answer: 3) 3

In a function, each input can have one and only one output. If k = 3, then the relation would contain (3,6) and (3,4), and would fail the vertical line test.

When you enter 3 into a function, it can't sometimes have 6 as its output and sometimes have 4 as its output.

More to come. Comments and questions welcome.

More Regents problems.

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