Tuesday, July 27, 2021

Math!

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Math * (Math - 1) * (Math - 2) * ... * 1

It depends on the value of Math. And some people give Math more value than others!



I also write Fiction!


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Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


16. The rectangular prism shown below has a length of 3.0 cm, a width of 2.2 cm, and a height of 7.5 cm.

What is the surface area, in square centimeters?

1) 45.6
2) 49.5
3) 78.0
4) 91.2

Answer: 4) 91.2


The surface area is the sum of the areas of the six sides of the prism. Since it is a rectangular prism, the opposite sides have the same area.

You have find the L * W, L * H, and W * H, double each of those, and find the sum.

L * W = 3 * 2.2 = 6.6

L * H = 3 * 7.5 = 22.5

W * H = 2.2 * 7.5 = 16.5

2 * (6.6 + 22.5 + 16.5) = 91.2





17. Which set of coordinates is a solution of the equation 2x - y = 11?

1) (-6, -1)
2) (-1, 9)
3) (0, 11)
4) (2, -7)

Answer: 4) (2, -7)


You can try each of the choices to see which makes a true statement, or you can rewrite the equation, solving for y, and put that into your graphing calculator and look at the Table of Values for the correct entry.

2x - y = 11
2x - 11 = y, or y = 2x - 11

Ironically, it wasn't until I typed the above equation just now that I noticed that the first three choices are all solutions to y = 2x + 11. I would've just did the mental math.





18. The graph of a parabola is represented by the equation y = ax2 where a is a positive integer. If a is multiplied by 2, the new parabola will become

1) narrower and open downward
2) narrower and open upward
3) wider and open downward
4) wider and open upward

Answer: 2) narrower and open upward


Since a is positive, it opens upward. Eliminate Choices (1) and (3).

If you double a, then the y values will increase more rapidly. That will cause the parabola to become thinner, or narrower.





19. Which equation represents a line that has a slope of 3/4 and passes through the point (2,1)?

1) 3y = 4x - 5
2) 3y = 4x + 2
3) 4y = 3x - 2
4) 4y = 3x + 5

Answer: 3) 4y = 3x - 2


The choices given are not in any familar form. However, they could easily be put into slope-intercept form by dividing by the coefficient of y to get y = mx + b.

If you divide Choices (1) and (2) by 3, then the value of m will be 4/3 instead of 3/4. Eliminate choices (1) and (2).

Next, substitute x = 2 into Choices (3) and (4).

4y = 3(2) - 2 = 6 - 2 = 4; 4y = 4, y = 1. This is the solution we are looking for.

4y = 3(2) + 5 = 6 + 5 = 11; 4y = 11, y = 11/4. This is NOT the solution we are looking for.





20. What is the value of | ( 4(-6) + 18 ) / 4! | ?
1) 1/4
2) - 1/4
3) 12
4) -12

Answer: 1) 1/4


I hate questions like this. I had a student come up to me after the exam and ask me, "Mr. Burke, what does an exclamation point in a math problem mean?" That's not to say that he shouldn't have known it, but the notation wasn't used as much as the concept it stands for. However, the reason I hate this problem is because it's meaningless and bizarre. There is no reason to put a factorial symbol in an equation like this other than to see if students were paying attention. It's a meaningless Order of Operations question with an added twist to it that only makes it longer, but not more challenging or interesting.

Order of Operations: Evaluate the numerator, multiply and add. Then evaluate the factorial, 4!, which is 4*3*2*1. Then simplify the fraction. Finally, take the absolute value.

Note that the Absolute Value is the final step, so the answer cannot be negative. Eliminate Choices (2) and (4).

| ( 4(-6) + 18 ) / 4! |
= | ( -24 + 18 ) / 4! |
= | -6 / 4! |
= | - 6/24 |
= | -1/4 |
= 1/4

I'm a little curious where they got 12 from. That should come from an "obvious" mistake, but I'm not seeing it.




More to come. Comments and questions welcome.

More Regents problems.

Monday, July 26, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


11. Is the equation A 21000(1 - 0.12)t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period?
1) exponential growth and 12%
2) exponential growth and 88%
3) exponential decay and 12%
4) exponential decay and 88%

Answer: 3) exponential decay and 12%


There is 12% decay, which will leave you with 88% of the original amount.

You know it is decay because of the minus sign, telling you that you will have less than 100% (the full amount) after the first time period.





12. The length of a rectangle is 15 and its width is w. The perimeter of the rectangle is, at most, 50. Which inequality can be used to find the longest possible width?
1) 30 + 2w < 50
2) 30 + 2w < 50
3) 30 + 2w > 50
4) 30 + 2w > 50

Answer: 2) 30 + 2w < 50


We are looking for the longest, meaning that there is a maximum number. All of the possibilities have to be LESS THAN that amount. Eliminate Choices (3) and (4).

The words "at most" 50 tell you that it can equal 50, so you want LESS THAN OR EQUAL TO.





13. Craig sees an advertisement for a car in a newspaper. Which information would not be classified as quantitative?
1) the cost of the car
2) the car's mileage
3) the model of the car
4) the weight of the car

Answer: 3) the model of the car


If you were unsure about quantitative vs qualitative, then this would be a good time to play "One of these things is NOT like the others".

Three of the four choices are numerical data and things that can be measured in dollars, miles and pounds. One of them is an arbitrary name given to the car, which is a description for the car.





14. What are the coordinates of the vertex and the equation of the axis of symmetry of the parabola shown in the graph below?


1) (0,2) and y = 2
2) (0,2) and x = 2
3) (-2, 6) and y = -2
4) (-2, 6) and x = -2

Answer: 4) (-2, 6) and x = -2


The vertex of a parabola is the turning point. It is either the maximum or the minimum point (and a parabola will only have one of those). In this example, the topmost point occurs at (-2, 6).

The axis of symmetry is a vertical line that, if drawn, would cut the parabola in half, creating two mirror images. The axis goes through the vertex.

The equation for all vertical lines is x = a, and every point on that line will be (a, y) for whatever value of y. The x coordinate of the vertex is -2.





15. A correct translation of “six less than twice the value of x” is
1) 2x < 6
2) 2x - 6
3) 6 < 2x
4) 6 - 2x

Answer: 2) 2x - 6


Do NOT confuse "less than" (subtraction) with "IS less than" (inequality).

Eliminate Choices (1) and (3) for using the < symbol instead of subtraction.

If my height is three inches less than your height, and your height is 68 inches, would you find my height by doing 68 - 3 or 3 - 68? The latter choice is ridiculous because it would give me a negative height.

"Six less than" something means "take six away from" something, and the something has to be there first to take something away from it.




More to come. Comments and questions welcome.

More Regents problems.

Sunday, July 25, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


6. Jason’s part-time job pays him $155 a week. If he has already saved $375, what is the minimum number of weeks he needs to work in order to have enough money to buy a dirt bike for $900?
1) 8
2) 9
3) 3
4) 4

Answer: 4) 4


You can set up an equation, or you can start with 375 and keeping adding 155 to it until you are over 900.

Set up an inequality in the form ax + b > c. The b represents the starting amount that he has, which is 375. This isn't repeated, so it doesn't get a variable. The a is the rate of change (the slope), which is the amount that occurs repeated, which is why it goes before the x. This is the 155.

155x + 375 > 900
155x > 525
x > 3.387...

Do NOT round down. He would have enough money after only three weeks. The first integer GREATER THAN 3.387... is 4. Jason will have money left over.





7. The expression 9a2 - 64b2 is equivalent to
1) (9a - 8b)(a + 8b)
2) (9a - 8b)(a - 8b)
3) (3a - 8b)(3a + 8b)
4) (3a - 8b)(3a - 8b)

Answer: 3) (3a - 8b)(3a + 8b)


The expression 9a2 - 64b2 is the difference of two perfect squares. Each part of it is a perfect square (which means you can take a square root of each component) and the two terms are separated by a minus sign.

When we have this, it means that the polynomial can be factored into two conjugates. Conjugates are two binomials that are identical, except that one has a plus and one has a minus separating them. When you multiply the two conjugates, you will get a zero pair with the two "middle" terms. (That is, the "OI" in "FOIL" will cancel out and become zero.)

9a2 - 64b2 = (3a - 8b)(3a + 8b). Note that the order of the two binomials doesn't matter.

Note that if you have two minus signs, then the middle terms, when combined, will not have a sum of zero.





8. The scatter plot below shows the profit, by month, for a new company for the first year of operation. Kate drew a line of best fit, as shown in the diagram.

Using this line, what is the best estimate for profit in the 18th month?
1) $35,000
2) $37,750
3) $42,500
4) $45,000

Answer: 3) $42,500


Interesting that there were two scatter plot questions so close together. (If you missed the previous post, question 4 was about correlation.)

Look at the line of best fit. Then go to 18 on the x-axis and move up to that line. You will see that the y-coordinate is between 40,000 and 45,000. So Choice 3 is the best estimate.





9. Which statement illustrates the additive identity property?
1) 6 + 0 = 6
2) -6 + 6 = 0
3) 4(6 + 3) = 4(6) + 4(3)
4) (4 + 6) + 3 = 4 + (6 + 3)

Answer: 1) 6 + 0 = 6


The identity property for addition states that adding 0 to a number doesn't change the number. It stays the same. (Multiplying by 1 for multiplication identity.)

Choice (2) shows an example of the Inverse Property of Addition, where adding two additive inverses yields a sum of zero. These two numbers are a zero pair.

Choice (3) is the Distributive Property of Multiplication over Addition.

Choice (4) is the Associative Property.





10. Peter walked 8,900 feet from home to school.

1 mile 5,280 feet

How far, to the nearest tenth of a mile, did he walk?
1) 0.5
2) 0.6
3) 1.6
4) 1.7

Answer: 4) 1.7


To change feet into miles, divide by 5,280.

8900 / 5200 = 1.71...

The sum of the leading coefficients 3 and -1 is 2, so eliminate Choice (4).

Since 8900 is greater than 5280, it's obvious that he walked more than a mile, so Choices (1) and (2) should be eliminated immediately.




More to come. Comments and questions welcome.

More Regents problems.

Friday, July 23, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


1. A system of equations is graphed on the set of axes below.


The solution of this system is
1) (0, 4)
2) (2, 4)
3) (4, 2)
4) (8, 0)

Answer: 3) (4, 2)


The solution to a system of equations is the point where the two lines intersect. That point is (4, 2). If you thought it was (2, 4), then you need to review which is the x coordinate and which is the y coordinate. Hint: the x comes first.





2. A cell phone can receive 120 messages per minute. At this rate, how many messages can the phone receive in 150 seconds?
1) 48
2) 75
3) 300
4) 18,000

Answer: 3) 300


Change minutes to seconds by dividing by 60 and then multiplying by 150 seconds.

120 / 60 * 150 = 300

If you multiplied by 60 and then divided by 150, you would've gotten 48. If you did 60 times 150 and then divided by 120 (I don't know why you would do that), you would get 75. If you just multiplied 120 by 150, you would have gotten 18,000. (It's rarely that easy when there's so much going on in a problem.)





3. The value of y in the equation 0.06y + 200 = 0.03y + 350 is
1) 500
2) 1,666.666...
3) 5,000
4) 18,333.333...

Answer: 3) 5,000


Note: On the actual exam, they used the repeater bar to indicate the repeating decimal.

0.06y + 200 = 0.03y + 350
0.03y + 200 = 350
0.03y = 150
y = 150/0.03 = 5,000

You can check your answer by substituting 5000 for y:

0.06(5000) + 200 = 0.03(5000) + 350
500 = 500 (check)





4. The scatter plot shown below represents a relationship between x and y.


This type of relationship is
1) a positive correlation
2) a negative correlation
3) a zero correlation
4) not able to be determined

Answer: 1) a positive correlation


First of all, we wouldn't say a "zero correlation". We'd say that there was no correlation. Is this splitting hairs? Maybe, but consider the difference between a zero slope and no slope. Words have meanings.

As the x values increase, the y values generally increase. So it is a positive correlation.





5. The sum of 3x2 + 5x - 6 and -x2 + 3x + 9 is
1) 2x2 + 8x - 15
2) 2x2 + 8x + 3
3) 2x4 + 8x2 + 3
4) 4x2 + 2x - 15

Answer: 2) 2x2 + 8x + 3


When you add polynomials, you combine like terms. The variables and their exponents do not change. Eliminate Choice (3).

The sum of the leading coefficients 3 and -1 is 2, so eliminate Choice (4).

5x + 3x = 8x and -6 + 9 = +3, so Choice (2) is the answer.

As the x values increase, the y values generally increase. So it is a positive correlation.




More to come. Comments and questions welcome.

More Regents problems.

Olympic Venn-ue

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Games, Sets, Matches

Not that we play with matches...



I also write Fiction!


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.

Thank you.





Come back often for more funny math and geeky comics.



Tuesday, July 20, 2021

Math+

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

More math content would be A+, right?

I could crack a few jokes here, or discuss possibilities. Or I could save that for future comics.



I also write Fiction!


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.

Thank you.





Come back often for more funny math and geeky comics.



Monday, July 19, 2021

School Life #22: Getting Closer

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Sometimes you need to retreat and regroup before you face the next level.

One for the Hal-Daisy fans.



I also write Fiction!


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.

Thank you.





Come back often for more funny math and geeky comics.



Friday, July 16, 2021

Argument of a Complex Number

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

They're not out of line here. Well, maybe above the line.

That's one problem with Algebra: there's always a lot of arguments!

The mathematical definition of Argument of a Complex Number is left as an exercise for the reader. But it's 3π/4 in this example.

It wasn't my intention to take a week off. It just worked out that way between life and the heat. Likewise, I considered doing something special for Comic #1750 (if I haven't started misnumbering again), but I settled for not doing a "mini" or a "quickie".

Final note: some of you might remember Negative Kitt from the other universe that was Through a Window, Darkly.

I also write Fiction!


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.

Thank you.





Come back often for more funny math and geeky comics.



Friday, July 09, 2021

Coordinate Hustlers

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Those who will not learn from Geometry will lose a lot of money to Geometry.

And as improbable as it may seem, whichever points are selected, the altitude in this problem will also be 3. The base AB is shown to be 3, so the area will always be 4.5 of those square boxes.



I also write Fiction!


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.

Thank you.





Come back often for more funny math and geeky comics.



Thursday, July 08, 2021

June 2021 Algebra 1 Regents (v202), Part I (multiple choice)



This exam was adminstered in June 2021. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

Part II is located here.

June 2021 (v202)

Part I

Each correct answer will receive 2 credits.


1. A high school club is researching a tour package offered by the Island Kayak Company. The company charges $35 per person and $245 for the tour guide. Which function represents the total cost, C(x), of this kayak tour package for x club members?

(1) C(x) = 35x
(2) C(x) = 35x + 245
(3) C(x) = 35(x + 245)
(4) C(x) = 35 + (x + 245)

Answer: (2) C(x) = 35x + 245


$35 per person is the cost that is multiplied by the x number of people going on the tour. The $245 is paid one time before anyone has even signed up for the tour, which makes it the y-intercept.





2. The expression 3(x + 4) - (2x + 7) is equivalent to

(1) x + 5
(2) x - 10
(3) x - 3
(4) x + 11

Answer: (1) x + 5


Distribute the 3 and distribute the minus sign (-1).
3(x + 4) - (2x + 7) = 3x + 12 - 2x - 7 = x + 5



3. A function is defined as K(x) = 2x2 - 5x + 3. The value of K(-3) is

(1) 54
(2) 36
(3) 0
(4) -18

Answer: (2) 36


K(-3) = 2(-3)2 - 5(-3) + 3 = 2(9) - (-15) + 3 = 18 + 15 + 3 = 36





4. Which relation is not a function?


Answer: (4)


In a function map, each member of the domain (on the left) can only have one arrow coming out of it because each input can only can associated with one output.

In Choice (4), there are two arrows coming out of the number six, meaning that (6, 10) and (6, 12) are in the same relation. This would cause it to fail the vertical line test.



5. The value of Tony's investment was $1140 on January 1st. On this date three years later, his investment was worth $1824. The average rate of change for this investment was $19 per

(1) day
(2) month
(3) quarter
(4) year

Answer: (4) -6


1824 - 1140 = 684, which is 684 / 3 = 228 per year, which is 228 / 12 = 19 per month.

This was an interesting question in that you had to work through the choices. However, number sense should tell you that "per month" was the most likely, so you could have tried 684 / 36 first.



6. The solution to 3(x - 8) + 4x = 8x + 4 is

(1) 12
(2) 28
(3) -12
(4) -28

Answer: (4) -28


Use the Distributive Property, and then solve the equation for x.
3(x - 8) + 4x = 8x + 4
3x - 24 + 4x = 8x + 4
7x - 24 = 8x + 4
-28 = x




7. An ice cream shop sells ice cream cones, c, and milkshakes, m. Each ice cream cone costs $1.50 and each milkshake costs $2.00. Donna has $19.00 to spend on ice cream cones and milkshakes. If she must buy 5 ice cream cones, which inequality could be used to determine the maximum number of milkshakes she can buy?

(1) 1.50(5) + 2.00m > 19.00
(2) 1.50(5) + 2.00m < 19.00
(3) l.50c + 2.00(5) > 19.00
(4) l.50c + 2.00(5) < 19.00

Answer: (1) 1.50(5) + 2.00m > 19.00


If she must buy 5 ice cream cones, then substitute c = 5, not m = 5. If she can't spend more than $19, than you need to have a < symbol.



8. When written in standard form, the product of (3 + x) and (2x - 5) is

(1) 3x - 2
(2) 2x2 + x - 15
(3) 2x2 - llx - 15
(4) 6x - 15 + 2x2 - 5x

Answer: (2) 2x2 + x - 15


First of all, you know that the answer will be a quadratic expression, so Choice (1) is out. And Choice (4) is NOT is standard form, so eliminate that.

(x + 3)(2x - 5) = 2x2 + 6x - 5x - 15 = 2x2 + x - 15



9. If x = 2, y = 3 SQRT(2), and w = 2 SQRT(8), which expression results in a rational number?

(1) x + y
(2) y - w
(3) (w)(y)
(4) y / x

Answer: (3) (w)(y)


The product of SQRT(2) and SQRT(8) is SQRT((2)(8)) = SQRT(16) = 4, which is a rational number. When that result is multiplied by 3 and 2, it will still be rational. (24)



10. Which product is equivalent to 4x2 - 3x - 27?

(1) (2x + 9)(2x - 3)
(2) (2x - 9)(2x + 3)
(3) (4x + 9)(x - 3)
(4) (4x - 9)(x + 3)

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


You can multiply these or graph them or factor the given expression.

If you "borrow & payback" then (4)(-27) = -108. What two factors of -108 have a sum of -3? Running through the list of factors will get you to (9)(-12). You have to "pay back" the 4.

Write out 4x2 - 12x + 9x - 27, and factor by grouping: 4x(x - 3) + 9(x - 3) = (4x + 9)(x - 3)



11. Given: f(x) = 2/3 x - 4 and g(x) = 1/4x + 1
Four statements about this system are written below.

I. f(4) = g(4)
II. When x = 12,f(x) = g(x).
III. The graphs of f(x) and g(x) intersect at (12,4).
IV. The graphs of f(x) and g(x) intersect at (4,12).
Which statement(s) are true?

(1) II, only
(2) IV, only
(3) I and IV
(4) II and III

Answer: (4) II and III


Statement I says that f(4) = 2/3(4) - 4 = -4/3 = g(4) = 1/4(4) + 1 = 2. This is not true.
Statement II says that f(12) = 2/3 (12) - 4 = 8 - 4 = 4 = g(12) = 1/4 (12) + 1 = 4. This is true.
Statement III restates statement II with extra information, which if that not only is f(12) = g(12) but they both equal 4.
Statement IV is not true because statement I is not true.



12. Which sketch represents the polynomial function f(x) = x(x + 6)(x + 3)?

Answer: Choice (1)


Choices (3) and (4) are not functions. They fail the vertical line test.

If f(x) = x(x + 6)(x + 3) then there are zeroes at x = 0, x + 6 = 0 and x + 3 = 0, or x = 0, x = -6 and x = -3.

That's Choice (1).



13. If the parent function of f(x) is p (x) = x2, then the graph of the function f(x) = (x - k)2 + 5, where k > 0, would be a shift of

(1) k units to the left and a move of 5 units up
(2) k units to the left and a move of 5 units down
(3) k units to the right and a move of 5 units up
(4) k units to the right and a move of 5 units down

Answer: (3) k units to the right and a move of 5 units up


Basic definition. You can try it in your graphing calculator by picking any value (e.g. 2) for k and observing how the graph changes.



14. Which expression is equivalent to (-4x2)3?

(1) -12x6
(2) -12x5
(3) -64x6
(4) -64x5

Answer: (3) -64x6


The exponent of 2 in (-4x2)3 goes only with the x. Everything in the parentheses gets the 3 exponent.

(-4)3 x2 * 3 = -64x6. Notice that when you take a power to another power, you have to multiply.





15. Which function has the smallest y-intercept?

Answer: Choice (1)


The y-intercept in Choice (1) is -6, as you can see from y = mx + b.

The y-intercept in Choice (2) is 1, as you can see from the row where x = 0.

The y-intercept in Choice (3) is -2, as you can see if you calculate f(0).

The y-intercept in Choice (4) is -2, as you can see on the graph at the y-axis.





16. Which domain would be the most appropriate to use for a function that compares the number of emails sent (x) to the amount of data used for a cell phone plan (y)?

(1) integers
(2) whole numbers
(3) rational numbers
(4) irrational numbers

Answer: (3) rational numbers


If you are comparing two whole numbers, you will have a ratio, and you would want rational numbers.



17. Eric deposits $500 in a bank account that pays 3.5% interest, compounded yearly. Which type of function should he use to determine how much money he will have in the account at the end of 10 years?

(1) linear
(2) quadratic
(3) absolute value
(4) exponential

Answer: (4) exponential


If it is compounded interest, then you need an exponential function. Simple interest is linear.



18. Given: the sequence 4, 7, 10, 13,. ..

When using the arithmetic sequence formula an = a1 + (n - 1)d to determine the 10th term, which variable would be replaced with the number 3?

(1) a1
(2) n
(3) an
(4) d

Answer: (4) d


The common difference, d, is 3 from term to term.



19. Below are two representations of data.
A: 2,5,5,6,6,6,7,8,9


Which statement about A and B is true?
(1) median of A > median of B
(2) range of A < range of B
(3) upper quartile of A < upper quartile of B
(4) lower quartile of A > lower quartile of B

Answer:


The medians of A and B are both 6, so eliminate Choice (1).

The range of A is 7 and the range of B is 6, so eliminate Choice (2).

The upper quartile of A is 7.5 and the upper quartile of B is 9. This is the correct statement.

The lower quartile of A and B are both 5.





20. Which system has the same solution as the system below?
x + 3y = 10
-2x - 2y = 4


(1) -x + y = 6, 2x + 6y = 20
(2) -x + y = 14, 2x + 6y = 20
(3) x + y = 6, 2x + 6y = 20
(4) x + y = 14, 2x + 6y = 20

Answer: (2) -x + y = 14, 2x + 6y = 20


This was an interesting question. Uusually when they ask something like this, the choices are multiples of the given equation (although 3 of the choices will not be multiplied correctly). This question, however, is just which two separate systems of equations happen to have the same solution, totally by coincidence.

Put these all in your graphing calulcator to find which ones intersect at the same point.

The given system has a solution of (-8, 6). Now you can graph the others, or you can plug in the values to see which gives you a true statement.

-(-8) + (6) = 14, which is =/= 6. Eliminate Choice (1).

-(8) + 6 = 14, (check), and 2(-8) + 6(6) = -16 + 36 = 20 (check). This is the solution.

If you wanted to solve the original system of equations algebraically


x + 3y = 10
-2x - 2y = 4

2x + 6y = 20
-2x - 2y = 4

4y = 24
y = 6
x + 3(6) = 10
x + 18 = 10 x = -8




21. Given the pattern below, which recursive formula represents the number of triangles in this sequence?


(1) y = 2x + 3
(2) y = 3x + 2
(3) a1 = 2, an = an - 1 + 3
(4) a1 = 3, an = an - 1 + 2

Answer: (4) a1 = 3, an = an - 1 + 2


Notice that Choices (1) and (2) are not recursive formulas, so both can be eliminated.

The first term has 3 triangles and each term after adds 2 more. This is Choice (4).



22. Students were asked to write an expression which had a leading coefficient of 3 and a constant term of -4. Which response is correct?

(1) 3 - 2x3 - 4x
(2) 7x3 - 3x5 - 4
(3) 4 - 7x + 3x3
(4) -4x2 + 3x4 - 4

Answer: (4) -4x2 + 3x4 - 4


Were my students to give me any of these answers, I'd have a problem. None of these is in standard form, which I might not care about in some instances, but if I'm asking for a specific leading coefficient, write it first!

The leading coefficient of (1) is 2. The leading coefficient of (2) is -3. The leading coefficient of (3) is 3, but the constant is 4, not -4. The leading coefficient of (4) is 3, and the constant is -4.



23. Sarah travels on her bicycle at a speed of 22.7 miles per hour. What is Sarah's approximate speed, in kilometers per minute?

(1) 0.2
(2) 0.6
(3) 36.5
(4) 36.6

Answer: (2) 0.6


Common sense should tell you that Choices (3) and (4) might be approximate answers in kilometers per hour, not minute.

To change from miles to kilometers, multiply by 1.609 (as per the back of the booklet). To change hours to minutes, divide by 60.

22.7 * 1.609 / 60 = 0.608..., which is approximately 0.6.



24. Which ordered pair does not fall on the line formed by the other three?

(1) (16,18)
(2) (12,12)
(3) (9,10)
(4) (3,6)

Answer:


Another interesting problem in that you are required to look at all for points to see with one doesn't belong.

The easiest thing to do is plot the four points on graph paper and see which one doesn't line up.

If you checked the slopes, you would have seen that lines from (16, 18) to the other three all would have different slopes.

(18 - 12) / (16 - 12) = 6/4 or 3/2, (18 - 10) / (16 - 9) = 8/7, (18 - 6) / (16 - 3) = 12 / 13. Since all three are different, you can be sure that (16, 18) is the odd point out.








More to come. Comments and questions welcome.

More Regents problems.

Wednesday, July 07, 2021

Why Did I -- ?

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