The following are some of the multiple questions from the recent January 2020 New York State Common Core Algebra I Regents exam.

Omitted images will be added soon.

### January 2020 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.

**1.** *If f(x) = 2(3*^{x}) + 1, what is the value of f(2)?

**Answer: (2) 19 **

f(2) = 2(3^{2}) + 1 = 2(9) + 1 = 18 + 1 = 19

**2.** * A high school sponsored a badminton tournament. After each round, one-half of the players were eliminated. If there were 64 players at the start of the tournament, which equation models the number of players left after 3 rounds?
*

**Answer: (1) y = 64(1 - .5)**^{3}

If you are eliminated half, then you are removing 50% or .5 of the teams. The number of teams is getting smaller, so eliminate the choices with "+". The exponent refers to the number of rounds.

This is an easy one to check: 64, 32, 16, 8. There are 8 teams after three rounds.

64(1 - .5)^{3} = 64(.5)^{3} = 64(.125) = 8

**3.** *Given 7x + 2 *__>__ 58, which number is not in the solution set?

**Answer: (1) 6 **

Note that for questions like this, when it is a linear inequality, not quadratic or something quirky, the answer is usually the smallest or the greatest value of the choices. You can check them first by substitution.

7(6) + 2 __>__ 58

42 + 2 __>__ 58

44 __>__ 58

This is true. If you substitute 8, you will get exactly 58, which is equal, and the other two numbers will be greater.

7x + 2 __>__ 58

7x __>__ 56

x __>__ 8

**4.** * Which table could represent a function?
*

**Answer: (2) **

If a value repeats in the x column with different values in the y column, it cannot be a function. It will fail the vertical line test. One input cannot have multiple outputs in a function.

**5.** *Which value of x make (x - 3) / 4 + 2 /3 = 17 / 12 true?
*

**Answer: (2) 6**

You can plug choices in. You can find a common denominator between 3 and 4 and add the fractions. Or you can multiply the entire equation by 12 to get rid of all the fractions.

Note: If this doesn't look right, I'll replace it with an image as soon as I can.

(12)[(x - 3) / 4 + 2 /3] = [17 / 12](12)

3(x - 3) + 4(2) = 17

3x - 9 + 8 = 17

3x - 1 = 17

3x = 18

x = 6
**6.** *Which expression is equivalent to 18x*^{2} - 50?

**Answer: (3) 2(3x - 5)(3x + 5) **

When you have a binomial like this, the first thing you check for is "Difference of Two Squares". Neither 18 nor 50 are perfect squares. However, if you factor them by 2, you get 2(9x^{2} - 25), and that binomial *is* the difference of two squares.

That means it can be factored into two **conjugates**, which will be the same two terms, one separated by a plus, and one separated by a minus: 2(3x - 5)(3x + 5).

**7.** *The functions f(x) = x*^{2} - 6x + 9 and g(x) = f(x) +k are graphed below.

*
*

Which value of k would result in the graph of g(x)?
**Answer: (4) -2 **

Adding k to a function will translate the function up or down. Since the vertex of f(x) is at (3,0) and the vertex of g(x) is at (3,-2), the value of k must be -2.

**8.** *The shaded boxes in the figure below represent a sequence.
*

*
*

If figure 1 represents the first term and this pattern continues, how many shaded blocks will be in figure 35?
**Answer: (2) 148 **

The first term has 12 boxes shaded. Each term after that adds 4 more boxes.

To get from figure 1 to figure 35, you have to add four 34 times.

So Figure 35 will have 12 + (34)(4) = 148 boxes.

**9.** *The zeros of the function f(x) = x*^{3} - 9x^{2} are

**Answer: (2) 0 and 9**

There is no constant term in the function, so if x = 0, then f(x) = 0. Choice (1) is incorrect.

You can work backward from the choices to see that 9 is a solution and that 3 and -3 are not.

Or you can solve x^{3} - 9x^{2} = 0

x^{2}(x - 9) = 0

x = 0 or x = 9

At no point is this a Difference of Squares problem.

**10.** * A middle school conducted a survey of student to determine if they spent more of their time playing games or watching videos on their tablets. The results are shown in the table below.
*

*
*

Of the students who spent more time playing games on their tablets, approximately what percent were boys?
**Answer: (3) 72 **

There were 138 boys and a total of 192 people in the Playing Games column. Turn that into a percent.

138 / 192 = 0.71875, or 72%

**11.** * Which statement best describes the solutions of a two-variable equation?
*

**Answer: (1) The ordered pairs must lie on the graphed equation. **

This is by design. You graph the line to find the solutions.

If the points are "near", they are "almost" correct, but still not correct.

There will be times when you want to find a point on the x-axis or y-axis, but those are not the only solutions to an equation.

**12.** *The expression x*^{2} - 10x + 24 is equivalent to

**Answer: (4) (x - 6)(x + 4)**

This is a popular question because either of the signs could be changed and you would still have the same four choices (only different answers).

The + 24 tells us the signs will be the same. The - 10x tells us that both of those signs will be negative.

(-6)(-4) = + 24. (-6x) + (-4x) = -10x

**13.** *Which statement is true about the functions f(x) and g(x), given below?
*

*
*
**Answer: (2) f(x) and g(x) have the same y-intercept **

If you read all the choices first, this one would have been the easiest to check, and it was the correct one.

f(0) = -4 and the graph of g(x) has y-intercept of -4.

Checking the others:

(3) f(x) and g(x) have the same roots. f(-2) = -(-2)^{2} - 4(-2) - 4 = 0, check; f(2) = -(2)^{2} - 4(2) - 4 = -16, does not check. Not Choice (3)

(1) The minimum value of g(x) is greater than the maximum value of f(x). False -- we just found a zero of f(x), which is greater than the minimum of g(x), which is -4.

(4) f(x) = g(x), when x = -4. f(-4) = -(-4)^{2} - 4(-4) - 4 = -16 + 16 - 4 = -4. (-4, -4) is not a point on g(x). False.

**14.** * The equation V(t) = 12,000(0.75)*^{t} represents the value of a motorcycle t years after it was purchased. Which statement is true?

**Answer: (2) The motorcycle cost $12,000 when purchased. **

The 12,000 represents the initial cost, the 0.75 represents the 25% depreciation, and the t is the number of years.

**15.** *The solutions to (x + 4)*^{2} - 2 = 7 are

**Answer: (3) -1 and -7 **

You can substitute the integer choices to see what works. Thankfully, one of them is the answer.

You can also graph y = (x + 4)^{2} - 9 and look at the zeros on the calculator.

Or you can solve it:

(x + 4)^{2} - 2 = 7

(x + 4)^{2} = 9

(x + 4) = __+__3

x = -3 - 4 = -7 or x = 3 - 4 = -1.
**16.** *Which expression is not equivalent to -4x*^{3} + x^{2} - 6x + 8?

**Answer: (2) (x)(-4x**^{2} - x + 6) + 8

In Choice (2), one x has been factored out of the first three terms and the last term has been left alone. However, two of the signs have been changed which should not have.

If you multiply/distribute the other choices, you will get the original expression back.

**17.** *Which situation could be modeled as a linear equation?
*

**Answer: (3) Two liters of water evaporate from a pool every day.**

It evaporates at the same rate, the same amount per day.

The others show exponential growth or decay.

**18.** *The range of the function f(x) = |x + 3| - 5 is
*

**Answer: (1) [-5, infinity) **

The vertex of the equation is (-3, -5). This is the minimum point. It opens upwards, and goes forever.

So the range -- the set of y-values -- goes from -5 (inclusive) to infinity.

Note: in the function *f(x) = a|x - h| + k*, (h, k) is the turning point. It's a minimum if a > 0. (In the above case, a = 1 and is understood to be there.)

**19.** *A laboratory technician used the function t(,) = 2(3)*^{2m + 1} to model her research. Consider the following expressions:

*I. 6(3)*^{2m}

II. 6(6)^{2m}

III. 6(9)^{m}
*
*

The function t(m) is equivalent to
**Answer: (3) I and III **

It should be obvious that there is no way that I and II could possibly get the same. The base changed, but the exponent did not, nor did the multiplier in front.

There rules of exponents say that when you multiply the same base, you add the exponents

So 2(3)^{2m + 1} is the same as 2(3)^{m}(3)^{m}(3)^{1}.

2(3)^{m}(3)^{m}(3) = 2(3) * (3)^{m} * (3)^{m} = 6(3)^{2m}

Also 6(3)^{2m} can be rewritten as 6((3)^{2})^{m}, which is 6(9)^{m}.

**20.** * Which system of equations has the same solutions as the system below?
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3x - y = 7

2x + 3y = 12*
*
**Answer: (2) 18x - 6y = 42; 4x + 6y = 24 **

The first equation was multiplied by 6 and the second by 2.

In Choice (1), the second equation is incorrect. The right side should be -36.

In Choice (3), the first equation should have a + 3y term, not a - 3y.

In Choice (4), there is no way to reduce 2x + 3y = 12 to x + y = 2. The slopes are different. The intercepts are different. It's just incorrect.

**21.** *A population of paramecia, P, can be modeled using the exponential function P(t) = 3(2)*^{t}, where t is the number o fdays since the population was first observed.Which domain is most appropriate to use to determine the populations over the course of the first two weeks?

**Answer: (4) 0 **__<__ t __<__ 14

The first two weeks is 14 days -- t is measured in days, not weeks.

Choices (1) and (2) are immediately incorrect because we have a definite beginning and definite end.

**22.** *Given the following data set:
*

65, 70, 70, 70, 80, 80, 80, 85, 90, 90, 95, 95, 95, 100*
*

Which representations are correct for this data set?
**Answer: (4) I, II, and III**

Check I, the line plot, and you will see each point matches up to one element of the data set.

Look at III, the frequency table. It has an interval of 10. If you mark off the data set at every ten, you will see that it matches up. Or you can circle pairs of columns on the line plot -- there are 5 dots in 65-70 and 61-70 has a frequency of 5, etc.

To check II, you might want to use the statistics functions on your graphing calculator to create a box and whisker plot, or to at least get a Five Number Summary.

There are 15 numbers in the data set, so the 8th one is the median, 80. Cross this out and there are seven below and seven above. The fourth number, 70, is the median for the lower half, which means it is the lower quartile, or Q1. Do the same for the upper half of the data, where the median is 95. This is the upper quartile, or Q3. The minimum is 65, and the maximum is 100.

The box and whisker plot shown has a five number summary of 65, 70, 80, 95, 100. It matches.

All three are correct representations for the data set.

**23.** *A recursively defined sequence is show below.
*

a_{1} = 5

a_{n + 1} = 2a_{n} - 7 *
*

The value of a_{4} is
**Answer: (1) -9 **

a_{1} = 5

a_{2} = 2(5) - 7 = 3

a_{3} = 2(3) - 7 = -1

a_{4} = 2(-1) - 7 = -9

**24.** *Which polynomial has a leading coefficient of 4 and a degree of 3?
*

**Answer: (4) 2x + x**^{2} + 4x^{3}

Leading coefficient is the first number when the expression is in standard form with the term with the highest exponent written first. The degree is the highest exponent.

So you are looking for an expression with 4x^{3} in it and no higher exponents.

Choices (1) and (3) have exponents of 4. Choice (2) has a leading coefficient of 5.

**End of Part I**

How did you do?

Questions, comments and corrections welcome.