The following are some of the multiple questions from the recent August 2019 New York State Common Core Algebra I Regents exam.

Omitted images will be added soon.

### August 2019 Algebra I, Part I

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

**1.** *Bryan’s hockey team is purchasing jerseys. The company charges $250
for a onetime set-up fee and $23 for each printed jersey. Which
expression represents the total cost of x number of jerseys for the
team?
*

**Answer: (3) 23x + 250 **

The start-up fee is paid once, so it doesn't get a variable. It's the cost even if 0 jerseys are made, so it's the y-intercept of a graph. The $23 is a repeated cost *per jersey*, so it gets a variable.

**2.** * Which table represents a function?
*

**Answer: (4)
**

Choices (1), (2) and (3) each repeat an x value with a different y value. That is not allowed in a function.

**3.** *Which expression is equivalent to 2(x ^{2} - 1) + 3x(x - 4)?
*

**Answer: (4) 5x ^{2} - 12x - 2 **

2(x

^{2}- 1) + 3x(x - 4)

2x

^{2}- 2 + 3x

^{2}- 12x

5x

^{2}- 12x - 2

**4.** * The value of x that satisfies the equation 4/3 = (x + 10) / 15 is
*

**Answer: (3) 10 **

The quick way: 3 goes into 15 five times, and 4 * 5 = 20 = x + 10. Then x = 10.

or

4/3 = (x + 10) / 15

(3)(x + 10) = (4)(15)

3x + 30 = 60

3x = 30

x = 10

**5.** * Josh graphed the function f(x) = -3(x - 1) ^{2} + 2. He then graphed the function g(x) = -3(x - 1)^{2} - 5 on the same coordinate plane.
The vertex of g(x) is
*

**Answer: (1) 7 units below the vertex of f(x)**

The vertex of f(x) is (1, 2). The vertex of g(x) is (1, -5).

Therefore the vertex of g(x) is 7 units lower than f(x).

**6.** *A survey was given to 12th-grade students of West High School to
determine the location for the senior class trip. The results are shown
in the table below.
*

Niagara Falls | Darien Lake | New York City | |

Boys | 56 | 74 | 103 |

Girls | 71 | 92 | 88 |

To the nearest percent, what percent of the boys chose Niagara Falls?

To the nearest percent, what percent of the boys chose Niagara Falls?

**Answer: (2) 24 **

Sum the number of boys by adding the three responses: 56 + 74 + 103 = 233.

Divide the number for Niagara Falls by 233, and make it a percent: 56 / 233 * 100% = 24%

Note that you don't need to use any of the data for the girls.

**7.** *Which type of function is shown in the graph below?
*

**Answer: (2) exponential **

Basically, a definition question. Linear functions and absolute value functions are straight lines, not curves.

A square root bends the other way, with smaller increases in y and x gets larger.

**8.** *The expression 16x ^{2} - 81 is equivalent to
*

**Answer: (3) (4x - 9)(4x + 9) **

The Difference of Squares Rule: when you have the difference between two terms which are each perfect squares, take the square root of each term and write the conjugates. (That is, write the two terms twice, once with a plus between them, and once with a minus.)

**9.** *The owner of a landscaping business wants to know how much time,
on average, his workers spend mowing one lawn. Which is the most appropriate rate with which to calculate an answer to his question?
*

**Answer: (4) hours per lawn **

Time per lawn: time can be measured in hours and lawns in the unit measurement.

Lawns per day doesn't give the amount of time for one lawn.

**10.** * A ball is thrown into the air from the top of a building. The height,
h(t), of the ball above the ground t seconds after it is thrown can be modeled by h(t) = -16t ^{2} + 64t + 80. How many seconds after being thrown will the ball hit the ground?
*

**Answer: (1) 5 **

Put the equation in your graphing calculator, and check the table. It has zeroes at -1 and 5. Reject -1 because negative time makes no sense.

The long way:

-16t^{2} + 64t + 80 = 0 -- Divide by -16 to get smaller numbers.

t^{2} - 4t - 5 = 0

(t - 5)(t + 1) = 0

t = 5 or t = -1, reject -1

**11.** * Which equation is equivalent to y = x ^{2} + 24x - 18?
*

**Answer: (1) y = (x + 12) ^{2} - 162 **

You don't have to complete the square because choices are given.

Actually, you wouldn't have to complete the square even without the choices.

Put the graph into the calculator and look at the table. Where is the vertex? It is at (-12, -162).

You can also see that (-12, 126), (12, 162) and (12, 126) are not even points on the parabola.

Next method: the vertex is on the Axis of Symmetry, so x = -b / (2a) = -24/(2*1) = -12.

Eliminate choices (3) and (4).

Then y = (-12)^{2} + 24(-12) - 18 = -162.

**12.** *When (x)(x - 5)(2x + 3) is expressed as a polynomial in standard
form, which statement about the resulting polynomial is true?
*

**Answer: (2) The leading coefficient is 2.**

(x)(x)(2x) = 2x^{3}, which is be the leading term, with the highest exponent. This also means that the degree is 3, and not 2.

The constant will be (-5)(3) = -15, not 2. And there will be more than these 2 terms.

**13.** * The population of a city can be modeled by P(t) = 3810(1.0005) ^{7t
}, where P(t) is the population after t years. Which function is approximately equivalent to P(t)?
*

**Answer: (2) P(t) = 3810(1.0035) ^{t} **

(1.0005)

^{7t}= (1.0005

^{7})

^{t}.

1.0005 to the 7th power is approximately 1.0035.

The rest of the equation remains unchanged.

**14.** * The functions f(x) and g(x) are graphed on the set of axes below.
*

For which value of x if f(x) =/= g(x)?

For which value of x if f(x) =/= g(x)?

**Answer: (3) 3 **

The graphs intersect at x = -2, x = -1 and x = 2. At x = 3, g(3) = 0, but f(3) is off the top of the graph.

**15.** *What is the range of the box plot shown below?
*

**Answer: (1) 7 **

The range is 8 - 1 = 7.

The difference 6 - 4 (which is 2) would be the **Interquartile Range**, which is different.

**16.** *Which expression is not equivalent to 2x ^{2} + 10x + 12?
*

**Answer: (3) (2x + 3)(x + 4) **

Not to be a broken record, but you could put each of these into the graphing calculator and see if they give you the same graph. However, in this case, that would actually take more time.

Just looking at the choices, you should immediately see that (2) and (4) are equivalent to each other because (2x + 6) is the same as 2(x + 3). So neither of those in the answer.

Likewise, there is no way that both (1) and (3) could be equivalent: switching the 3 and 4 will result in a different middle term. So check those:

(2x + 4)(x + 3) = 2x^{2} + 6x + 4x + 12 =
2x^{2} + 10x + 12. Check!

(2x + 3)(x + 4) = 2x^{2} + 8x + 3x + 12 =
2x^{2} + 11x + 12. Wrong!

**17.** *The quadratic functions r(x) and q(x) are given below
*

x | r(x) |

-4 | -12 |

-3 | -15 |

-2 | -16 |

-1 | -15 |

0 | -12 |

1 | 7 |

q(x)=x^{2} + 2x - 8

*The function with the smallest minimum value is
*

**Answer: (3) r(x), and the value is -16 **

The minimum for r(x) is -16.

Graph q(x) and find the vertex. Or use the Axis of Symmetry, x = -2/(2(1)) = -1.

Evaluate q(-1) = (-1)^{2} + 2(-1) - 8 = 1 - 2 - 8 = -9.

The function with the smallest minimum value is r(x), and that value is -16 (which occurs when x = -2, but that is NOT what is being asked).

**18.** *A child is playing outside. The graph below shows the child's distance, d(t), in yards from home over a period of time, t, in seconds.
*

*Which interval represents the child constantly moving closer to home?
*

**Answer: (1) 0 < t < 2 **

If the child is moving toward home at a constant rate, the distance is decreasing steadily.

That happens between 0 and 2 seconds. at you're accidentally get the "right" answer to appear by mistake.

**19.** *If a _{1} = 6 and a_{n} = 3 + 2(a_{n - 1})^{2}, the a_{2} equals
*

**Answer: (1) 75 **

a_{2} = 3 + 2(a_{2 - 1})^{2} = 3 + 2(a_{1})^{2}

= 3 + 2(6)^{2} = 3 + 2(36) = 3 + 72 = 75

**20.** * The length of a rectangular patio is 7 feet more than its width, w. The
area of a patio, A(w), can be represented by the function
*

**Answer: (2) A(w) = w ^{2} + 7w **

Length = w + 7. Area = length * width = (w + 7)(w) = w

^{2}+ 7w.

**21.** *A dolphin jumps out of the water and then back into the water. His
jump could be graphed on a set of axes where x represents time and y represents distance above or below sea level. The domain for this graph is best represented using a set of
*

**Answer: (4) positive real numbers**

Forget about the above and below the water. They asked for the domain, which is the x value, which in this case represents time. Negative time makes no sense.

I don't like this problem because it leaves out 0, which is not positive, but should be part of the domain. However, the answer key tells me that it is not part of the domain.

**22.** *Which system of linear equations has the same solution as the one
shown below?
*

x - 4y = 10

x + y = 5

x - 4y = 10

x + y = 5

**Answer: (1) 5x = 10, x + y = 5 **

Change x + y = 5 into y = 5 - x

That makes the first equation x - 4(5 - x) = -10

and then x - 20 + 4x = -10

So 5x - 20 = -10

And 5x = 10, which is choice (1).

**23.** *Which interval represents the range of the function
h(x) = 2x ^{2} - 2x - 4?
*

**Answer: (4) [4.5, ∞) **

It's a parabola that opens upwards, so the upper boundary is infinity. The lower boundary is the minimum point at the vertex. Just looking at the choices, the answer should be obvious once you realize what the incorrect answer is.

You can graph this and look for the vertex.

Or use the Axis of Symmetry: x = -b/(2a) = -(-2)/(2(2)) = 0.5, which is NOT the range.

h(.5) = 2(.5)^{2} - 2(.5) - 4 = -4.5, which is part of the range, so use [ not (.

**24.** *What is a common ratio of the geometric sequence whose first term
is 5 and third term is 245?
*

**Answer: (1) 7 **

The second term must be n/5 = 245/n, so n^{2} = (5)(245)

n = SQRT(1225) = 35

35 / 5 = 7, and 245 / 35 = 7

The Common ratio is 7.

**End of Part I**

How did you do?

Questions, comments and corrections welcome.

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