June 2024 Algebra 2 Regents Part III

This exam was adminstered in January 2024 .

June 2024 Algebra 2, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

33. Solve the following system of equations algebraically for all values of x, y, and z:

3x - 8y + 2z = -60
2x - 7y - 5z = -31
-6x + 2y - 4z = 36

Answer:

You can solve a system of three equations with three variables by eliminating one or more variables through multiplication and elimination.

In this example, if you look at the first and third equations, you will see that -6x is 3x * -2 and -4z is 2x * -2, so if you multiply the first equation by 2, you can eliminate two variables at once.

3x - 8y + 2z = -60
-6x + 2y - 4z = 36

6x - 16y + 4z = -120
-6x + 2y - 4z = 36

-14y = -84
y = 6

Once you know that y = 6, you can substitute into the original equations and solve for the other two variables.

3x - 8(6) + 2z = -60 --> 3x + 2z = -12
2x - 7(6) - 5z = -31 --> 2x - 5z = 11
-6x + 2(6) - 4z = 36 --> -6x - 4z = 24

3x + 2z = -12
2x - 5z = 11
6x + 4z = -24
-6x + 15z = -33

19z = -57
z = -3

3x - 8(6) + 2(-3) = -60
3x - 48 - 6 = -60
3x = -6
x = -2

You can check your answers by substituting your solutions into the other equations and getting true statements.

34. In the town of Skaneateles, New York, house prices since 2008 have changed based on the function H(t) = 200,000(1.045)^t, where t is the number of years since 2008 and H(t) is the median house price. Determine the average rate of change for the median house price in Skaneateles from 2010 to 2018 to the nearest dollar per year.

Explain what this rate of change means as it relates to median house prices.

Answer:

THe average rate of change is the difference between H(10) and H(2) divided by 8 years (2018 - 2010).

(200000(1.045)¹⁰ - 200000(1.045)²) / 8 = 11523.610...

To the nearest dollar, the average rate of change is $11524. (Make sure you round UP.)

This means that the median home price rose about $11524 every year from 2010 to 2018.

35. IA sporting goods manufacturer is trying to determine if they should continue to produce multiple types of hockey pucks. The company surveyed 50 randomly chosen customers and asked them if they purchased both game regulation pucks and lighter training pucks. Of those surveyed, 40 of them said that they purchase both types of pucks. A simulation that was run 100 times based on the survey results produced the approximately normal results below

a) Determine an interval containing the middle 95% of plausible values that estimates the proportion of all customers who would purchase both types of pucks from the company.

b) The company will continue to manufacture both types of hockey pucks if it is reasonable to assume that the true proportion of customers who buy both types of hockey pucks is above 0.60. Using the interval from part a, explain whether or not the company should continue to produce both types of hockey pucks.

Answer:

The middle 95% of the data will fall within two Standard Deviations (SD) from the mean. This will leave 2.5% of the data on the low end and 2.5% of the data on the high end. The Mean is 0.795 and the SD is 0.085.

This means that 95% of the data falls within the range:

0.795 - 2(0.085) = 0.625
0.795 + 2(0.085) = 0.965

The range is 0.625 - 0.965

The company should continue to make both types because 0.625, which is the bottom of the confidence interval, is greater than .60.

36. Graph y = f(x), where f(x) = log₂(x - 1) + 3 on the set of axes below.

State the equation of the asymptote of f(x).

When f(x) is reflected over the line y = x, a new function is formed: g(x) = 2^{(x - 3)} + 1. State the equation of the asymptote of g(x).

Answer:

Put the equation intop your graphing calculator and check the table of values. You will get the points (2,3), (3,4), (5,5), (9,6). You will see that there is a vertical asymptote at x = 1. You will see that the graph crosses the x-axis at a number a little more than x = 1 (at x = 1.125).

The asymptote is x = 1.

If the function is reflection over the line y = x, then the asymptote of g(x) will be y = 1 because g(x) is an exponential function, which would have an asymptote of y = 0 but there is a constant of +1 which raises it up to y = 1.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: I See What You'll Do There, written by Christopher J. Burke, which contains two humorous fantasy stories Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend and Portrait of a Lady Vampire are still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

June 2024 Algebra 2 Regents Part II

This exam was adminstered in June 2024 .

June 2024 Algebra 2, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Given x is a real number, write the expression in simplest a + bi form:

(x + 2i)(3 - 2xi) + 2x²i

Answer:

Multiply the binomials using FOIL or the Box Method, keeping in mind that i² = -1. Then combine like terms.

(x + 2i)(3 - 2xi) + 2x²i
3x - 2x²i + 6i - 4xi² + 2x²i
3x - 2x²i + 6i + 4x + 2x²i
3x + 6i + 4x
7x + 6i

26. Solve 3.8e^1.5t = 16 algebraically for t to the nearest hundredth.

Answer:

Inverse operations. Have a calculator handy.

3.8e^1.5t = 16
e^1.5t = 16/3.8
ln (e^1.5t) = ln(16/3.8)
1.5t = 1.43758765551
t = 0.95839177033

t = 0.96 to the nearest hundredth.

27. In an attempt to get the student body’s opinion of a new dress code, members of the statistics class surveyed the students of the first period computer science class. Explain a statistical bias in the method of data collection.

Answer:

There are several possible answers that would show bias.

The computer science class is not representative of how the entire student body dresses.

Only surveying first period students excludes students who have schedules that start later.

Other answers are possible, but basically, point out that this is not a random sample of students in the school.

28. Sketch a graph of polynomial P(x), given the criteria below:

• P(x) has zeros only at -5, 1, and 4
• As x → ∞, P(x) → -∞
• As x → -∞, P(x) → -∞

Answer:

The graph must start at the bottom and end at the bottom because it comes up from negative infinity and goes off to negative infinity. In between it has to cross or touch the x-axis at -5, 1, and 4. One of those must be a double root. That is, the curve will touch the x-axis and turn back instead of crossing the axis.

A possible solution looks like this:

It's just a sketch, so it doesn't have to be perfectly measured out. However, if you draw tick marks on the x-axis, then the zeros better be at -5, 1, and 4. You should have arrows on the ends of the lines to indicate that the graph continues in the direction shown. (Seriously, you'll lose a point without them because that would suggest endpoints to the graph.)

The turnaround point could happen at any of the three points but it has to be at one of them. It could also be at all three of them if it was a sixth-degree polynomial, P(x) = (x+5)²(x-1)²(x-4)².

29. The height, above ground, of a Ferris wheel car can be modeled by the function h(t) = - 103.5cos(2πt/5) + 108.5 where h is measured in feet and t is measured in minutes.
State the period of the function and describe what the period represents in this context.

Answer:
The period of cos(t) is 2π. The period of cos (2πt/5) is 2π / (2π/5), which is 5. The period is 5.

In this context, the period is how long it takes the Ferris wheel to complete one full revolution, which in this case is 5 minutes.

That sounds like a lot but it's a really big Ferris wheel with a radius of 103.5 feet. The center is 108.5 feet above the ground.

30. Solve algebraically for all values of x:

8 / (x + 5) - 3 / x = 5

Answer:
Multiply each term by (x) and (x + 5) to eliminate the denominators. This will result in a quadratic equation that needs to be solved.

8 / (x + 5) - 3 / x = 5

8(x)(x + 5) / (x + 5) - (3 (x)(x + 5)) / x = 5(x)(x + 5)

8x - 3x - 15 = 5x² + 25x

8x - 3x - 15 = 5x² + 25x

0 = 5x² + 20x + 15

0 = x² + 4x + 3

0 = (x + 3)(x + 1)

x + 3 = 0 or x + 1 = 0

x = -3 or x = -1

Neither solution causes a zero denominator, so both are accepted and neither is discarded.

31. The transportation methods used by the upperclassmen at Calhoun High School are summarized in the table below.

Are the events “being a junior” and “driving to school” independent? Using statistical evidence, justify your answer.

Answer:
If "being a junior" and "driving to school" are independent then the probability of being a junior who drives would equalt the probability of being a junior times the probability of driving.

If you add up the subtotals, you will see that there are 145 juniors and 139 upperclassmen who drive. There are a total of 277 upperclassmen.

P(J and D) = 58/277 = .209

P(J) * P(D) = 145/277 * 139/277 = .263

The probabilites are not the same. Therefore "being a junior" and "driving to school" are NOT independent events.

You must have a concluding statement. You can just write the two decimals without explainging what they mean.

32. Can f(x) = x³ + 7 be classified as an odd function? Justify your answer

Answer:
No, f(x) is neither an odd or an even function.

An odd function will only have odd exponents on the variables. However, the constant term (x⁰ term) has an exponent of zero, which is even.

Odd functions have rotational symmetry. That means that odd functions must go through the origin. However, f(0) = 7, not zero, so it can't be an odd function.

Finally, for a function to be odd f(-x) must equal -f(x). So f(-1) must equal -f(1) and f(0) must equal -f(0).

F(-1) = 6, but -f(1) = -8, and f(-0) = 7, but -f(0) = -7. So f(x) is not odd.

In general, f(-x) = (-x)³ + 7, but -f(x) = -(x³ + 7) = -(x³) - 7. so f(-x) =/= -f(x).

Either of the two explanations or the examples above are sufficient for full credit.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: I See What You'll Do There, written by Christopher J. Burke, which contains two humorous fantasy stories Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend and Portrait of a Lady Vampire are still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

Connecting the Dots

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Looks like they won't be forming any connections.

Welcome back to the comics. This is number 1982 (unless I missed a strip when I was counting recently), and I hope to make it to 2000.

If nothing else, I want to finish story lines that I started in May when my real-life class was so far behind with the new curriculum thrust upon us (and which I was rated "Developing" on curriculum design for a lesson I didn't design) that I had no time to work on anything more complex than the comic above. And as these things work out, the longer you go without getting back to you, the harder it is to get back to it.

By the end of June, I started thinking I should put out a book about Everything You Need to Know About Geometry ... but I remember how badly my last math book turned out. Granted, I'm getting better at it.

Any way ... let's see if I can get two comics out this week.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: I See What You'll Do There, written by Christopher J. Burke, which contains two humorous fantasy stories Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend and Portrait of a Lady Vampire are still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


17. The students in Mrs. Smith’s algebra class were asked to describe the graph of g(x) = 2(x - 3)² compared to the graph of f(x) = x².
Which student response is correct?

(1) Ashley said that the graph of g(x) is wider and shifted left 3 units.
(2) Beth said that the graph of g(x) is narrower and shifted left 3 units.
(3) Carl said that the graph of g(x) is wider and shifted right 3 units.
(4) Don said that the graph of g(x) is narrower and shifted right 3 units.

Answer: (4) Don said that the graph of g(x) is narrower and shifted right 3 units.

If the leading coefficient of a quadratic function is greather than 1, then the parabola will become skinnier, or narrower. (I like to say "skinnier" because I have trouble saying "narrower" in a classroom setting. I don't know why -- I just do.) Eliminate Choices (1) and (3) which happen when the coefficient is between 0 and 1.

If you subtract three from x inside the parentheses, the entire graph will shift to the right. So Choice (4) is the correct response.

18. One Saturday, Dave took a long bike ride. The graph below models his trip.

What was Dave’s average rate of change, in miles per hour, on this trip?

(1) 10
(2) 11
(3) 11.6
(4) 14.5

Answer: (1) 10

The average rate of change in miles per hour is literally the number of miles he traveled divided by the number of hours he traveled.

The final coordinate is (5.5, 55). Divide 55/5.5 = 10, which is Choice (1).

11. Which expression is equivalent to (x - 5)(2x + 7) - (x + 5)?

(1) 2x² - 2x - 30
(2) 2x² - 2x - 40
(3) 2x² - 4x - 30
(4) 2x² - 4x - 40

Answer: (4) 2x² - 4x - 40

If you multiply (-5)(7), you get -35, then subtract +5, you get -40. That means that Choices (1) and (3) can be eliminated.

(x - 5)(2x + 7) - (x + 5)
2x² + 7x - 10x - 35 - x - 5
2x² - 4x - 40

Choice (4) is the correct answer.

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

What is the solution to the equation f(x) = g(x)?

(1) 1 and 5
(2) -5 and 0
(3) -3 and 5
(4) 0 and 4

Answer: (1) 1 and 5

For which values of x will f(x) = g(x). They do NOT want the values of f(x) or g(x). They want x.

Look for the points on the graph where the two functions intersect. Do not look at the points where the lines cross the x-axis -- that's a different kind of problem.

Looking at the graph, the lines intersect at x = 1 and x = 5. This is Choice (1).

Choice (2) is the y-intercepts of the two graphs.

Choice (3) is the values of f(x) and g(x), which isn't the question.

Choice (4) is the roots of f(x) and has nothing to do with g(x).

21. When babysitting, Nicole charges an hourly rate and an additional charge for gas. She uses the function C(h) = 6h + 5 to determine how much to charge for babysitting. The constant term of this function represents

(1) the additional charge for gas
(2) the hourly rate Nicole charges
(3) the number of hours Nicole babysits
(4) the total Nicole earns from babysitting

Answer: (1) the additional charge for gas

The fee for gas is the constant, which is Choice (1).

The hourly rate is 6, which is the coefficient of the variable (the slope of the line, the rate of change).

The number of hours Nicole babysits is the variable.

The total Nicole earns is the value of C(h).

22. When solved for x in terms of a, the solution to the equation 3x - 7 = ax + 5 is

(1) 12 / (3a)
(2) 12 / (3-a)
(3) 3a / 12
(4) (3 - a) / 12

Answer: (2) 12 / (3-a)

Get all of the x terms on one side of the equation, and then isolate the variable.

3x - 7 = ax + 5
3x - ax = 5 + 7
x(3 - a) = 12
x = 12 / (3 - a)

This is Choice (2).

23. Wayde van Niekerk, a runner from South Africa, ran 400 meters in 43.03 seconds to set a world record. Which calculation would determine his average speed, in miles per hour?

Answer: (3) 400 m / 43.04 sec * 0.62 mi / 1000 m * 1 hr / 3600 sec

Units can be multipled and canceled like numerical factors can. When all the factors are canceled, only miles (on top) and hours (on the bottom) should remain.

In Choice (1), neither m nor sec cancel. It says m²hr / (mi * sec²). Eliminate Choice (1).

In Choice (2), sec doesn't cancel. It say mi * hr / (m * sec²). Eliminate Choice (2).

In Choice (3), both m and sec cancel, leaving mi/hr. This is the correct answer.

In Choice (4), m doesn't cancel. It say m² * sev / (sec * mi * hr). Eliminate Choice (4).

24. Which function has a domain of all real numbers and a range greater than or equal to three?

(1) f(x) = -x + 3
(2) g(x) = x² + 3
(3) h(x) = 3^x
(4) m(x) = |x + 3|

Answer: (2) mean = median

The domain of all these function is all real numbers. None of these functions are undefined for any value of x.

To have a range of greater than or equal to 3, no output of the funtion can be less than 3.

f(1) = -1 + 3 = 2. Eliminate Choice (1).

h(0) = 3⁰ = 1. Eliminate Choice (3).

m(-3) = |-3 + 3| = 0. Eliminate Choice (4).

g(x) = (x - 0)² + 3 have a vertex, the minimum point, at (0,3). The range must be 3 or greater. This is the correct answer.

Another way to look at it: Choice (1) is a linear equation that goes down to negative infinity. It has a range of all real numbers. Choice (2) is an exponential function that has an asymptote at y = 0 (the x-axis), so the range is greater than 0. Choice (4) is an absolute value function, so its range is greater than or equal to 0.

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. The zeros of the function f(x) = x(x - 5)(3x + 6)

(1) 0, -5, and 2
(2) 0, 5, -2
(3) -5 and 2, only
(4) 5 and -2, only

Answer: (2) 0, 5, -2

The zeroes are the values x that make the function equal to zero.

Since the first of the three terms that are being multiplied is just x, then the funtion is equal to 0 when x = 0, because zero times anything is zero. Eliminate Choices (3) and (4).

The second term (x - 5) is equal to 0 when x = 5 because (5 - 5) = 0. Eliminate Choice (1).

Choice (2) is the answer because (5 - 5) = 0 and (3(-2) + 6) = 0.

10. What is the y-intercept of the line that passes through the points (-1,5) and (2,-1)?

(1) -1
(2) -2
(3) 3
(4) 5

Answer: (3) 3

The y-intercept is the value of y when x = 0, or the y-coordinate when the line crosses the y-axis.

Before you do anything, it should be obvious that the number must be in-between 5 and -1, so eliminate Choices (1), (2), and (4).

Choice (3) is the only possible Choice and is the correct answer.

If you didn't realize this, you can use the scratch graph paper to plot the points and draw the line, using a straightedge or ruler, to find the answer.

You could also find the slope of the line: (-1 - 5) / (2 - -1) = -6 / 3 = -2.

If the slope is -2, then the point after (-1, 5) is (0, 3), which is Choice (3).

11. Nancy has just been hired for her first job. Her company gives her four choices for how she can collect her annual salary over the first eight years of employment.
Each function below represents the four choices she has for her annual salary in thousands of dollars, where t represents the number of years after she is hired.

a(t) = 2^t + 25
b(t) = 10t + 75
c(t) = √(400t) + 80
d(t) = 2(t + 1)² - 10t + 50
Which pay plan should Nancy choose in order to have the highest salary in her eighth year?

(1) a(t)
(2) b(t)
(3) c(t)
(4) d(t)

Answer: (1) a(t)

First, an important note to all high school students taking this exam: THIS WILL NEVER HAPPEN IN REAL LIFE!

They are asking which will be the highest amount a(8), b(8), c(8) or d(8). That's it. Forget about salaries and jobs. Just put them in the calculator, and be sure that you entered them correctly when it comes to the exponents and square roots.

a(8) = 2⁸ + 25 = 281

b(8) = 10(8) + 75 = 155

c(8) = √(400(8)) + 80 = 136.57

d(8) = 2(8 + 1)² - 10(8) + 50 = 132

The maximum amount is Choice (1), a(t), which you might have suspected because it is the only one that is exponential, and exponential functions rise more quickly than other functions. But it doesn't hurt to check.

12. The third term in a sequence is 25 and the fifth term is 625. Which number could be the common ratio of the sequence?

(1) 1/5
(2) 5
(3) 1/25
(4) 25

Answer: (2) 5

The sequence is increasing, so the common ratio must be greater than 1. Eliminate Choices (1) and (3).

If you multiply, 25 * 5 = 125 and 125 * 5 = 625, you have the correct answer. Choice (2) is the correct choice.

If you multiply 25 * 25 = 625, then the third term is 625, not the fourth term.

How you're supposed to solve this, if you didn't have choices to work with, is as follows:

25 * x is term 3, and term 3 * x = 625, so 25 x² = 625.
That means that x² = 25, and x = 5.

13. The box plot below summarizes the data for the amount of snowfall, in inches, during the winter of 2021 for 12 locations in western New York.

What is the interquartile range?

(1) 30
(2) 50
(3) 80
(4) 110

Answer: (2) 50

The interquartile range is the difference between the third or upper quartile (Q3) and first or lower quartile (Q1). Basically, it's the length of the box.

Q3 = 110, and Q1 = 60, and 110 - 60 = 50, so the IQR = 50, which is Choice (2).

Choice (1), 30, is the length from the median to Q3. Eliminated Choice (1).

Choice (3), 80, is the median of the data, not the length of the box. Eliminate Choice (3).

Choice (4), 110, is Q3, not the interquartile range. Eliminate Choice (4).

14. Four quadratic functions are represented below.

Which function has the smallest minimum value?

(1) I
(2) II
(3) III
(4) IV

Answer: (1) I

You can check the minimum value of function c(x) in your calculator, or you can work it out.

Function a(x) is in vertex form, meaning that (3,-7) is the vertex, so the minimum value for a(x) = -7. The minimum value of b(x) is -4 and the minumum value of d(x) is -5. Eliminate Choices (2) and (4).

The minimum value of c(c) is on the Axis of Symmetry, which is the line with the equation x = -b/(2a) = -(6)/(2) = -3. The value of c(-3) = (-3)² + 6(-3) + 3 = -6. Eliminate Choice (3).

Choice (1), I, is the correct answer.

15. The equation that represents the sequence -2, -5, -8, -11, -14, . . . is

(1) aⁿ = -3 + (-2)(n - 1)
(2) aⁿ = -2 + (-3)(n - 1)
(3) aⁿ = 3 + (-2)(n - 1)
(4) aⁿ = -2 + (3)(n - 1)

Answer: (2) aⁿ = -2 + (-3)(n - 1)

The initial term is -2 and the common difference is -3. The Choice that shows this is Choice (2), which is the Correct answer.

Check: Let's let n = 4 to find the value of the fourth term

-3 + (-2)(4 -1) = -9, not -11. Eliminate Choice (1).
-2 + (-3)(4 - 1) = -11. This is correct.
3 + (-2)(4 - 1) = -3, not -11. Eliminate Choice (3).
-2 + (3)(4 - 1) = 7, not -11. Eliminate Choice (4).

Only Choice (2) gives the correct fourth term.

16. The dot plot below shows the number of goals Jessica scored in each lacrosse game last season.

Which statement about the dot plot is correct?

(1) mean > mode
(2) mean = median
(3) mode = mediam
(4) median %gt; mean

Answer: (2) mean = median

The mode is 3, because there are 5 of them. The median is 2.5 because there are 20 pieces of data and the 10th is 2 and the 11th is 3, so the median is 2.5. The median does not equal the mode, so eliminate Choice (3).

The other three choices use the mean, so you have to calculat the mean, which is the average of all the data. That is, add up all 20 numbers and divide by 20. You can take a shortcut by using multiplication for each individual score. Obviously, I don't need to add the zeros, but I'll write it down just for completeness.

(3 * 0) + (3 * 1) + (4 * 2) + (5 * 3) + (2 * 4) + (2 * 5) + (1 * 6) = 50
50 / 20 = 2.5

The mean is not greater than the mode, so Choice (1) is eliminated.

The mean is 2.5 which is equal to the median. The correct answer is Choice (2).

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. A ball was launched into the air, and its height above the ground was recorded each second, as shown in the table below.

Based on these data, which statement is a valid conclusion?

(1) The ball lands on the ground at 4 seconds.
(2) The ball reaches a maximum height of 11 feet.
(3) The ball was launched from a height of 0 feet.
(4) The ball reaches its maximum height at 2 seconds.

Answer: (4) The ball reaches its maximum height at 2 seconds.

The ball starts at 11 feet from the ground, which is at 0 feet. Read the table to find the correct answer.

Choice (1) says the ball is on the ground at 4 seconds. It is not. It is at 11 feet. Eliminate Choice (1).

Choice (2) says the ball is reaches a maximum of 11 feet. No, it starts at 11 feet and goes up to 75 feet. Eliminate Choice (2).

Choice (3) says the ball was launched from 0 feet. No, it starts at 11 feet at 0 seconds. Eliminate Choice (3).

Choice (4) says the ball reaches its maximum height at 75 feet. The point (2,75) is the highest in the table. Also, the numbers on either side are symmetrical, so you can be sure that (2,75) is the vertex of the parabola that these points would graph. This is the correct answer.

2. A tour bus can seat, at most, 48 passengers. An adult ticket costs $18 and a child ticket costs $12. The bus company must collect at least $650 to make a profit. If a represents the number of adult tickets sold and c represents the number of child tickets sold, which system of inequalities models this situation if they make a profit?

(1) a + c < 48 ; 18a + 12c > 650
(2) a + c < 48 ; 18a + 12c > 650
(3) a + c < 48 ; 18a + 12c < 650
(4) a + c < 48 ; 18a + 12c < 650

Answer: (2) a + c < 48 ; 18a + 12c > 650

At most means "is less than or equal to". At least means "is greater than or equal to".

Immediately eliminate Choice (1) and (3) which don't have the "equal to" portion.

Choice (2) has the correct symbols that match the language in the question.

3. Which equation is always true?

(1) x² • x³ = x⁵
(2) 3^x • 3² = 9^2x
(3) -z² = z²
(4) 7^a • 7^b = 7^ab

Answer: (1) x² • x³ = x⁵

When you multiply two terms that have the same base, you keep the base and add the exponents.

In Choice (1), x² • x³ = x x • x x x = x⁵. This is the correct answer.

In Choice (2), 3^x • 3² = 3^(x+2), not 9^2x. Eliminate Choice (2).

In Choice (3), -z² = z² is not true for any value other that z = 0 because of the Order of Operations. What is true is (-z)² = z². Eliminate Choice (3).

In Choice (4), 7^a • 7^b = 7^a+b , not 7^ab. Eliminate Choice (4).

4. The expression -2(x² - 2x + 1) + (3x² + 3x - 5) is equivalent to

(1) x² + x - 4
(2) x² - x - 7
(3) x² + 7x - 4
(4) x² + 7x - 7

Answer: (4) x² + 7x - 7

Use the distributive property and then combine like terms.

-2(x² - 2x + 1) + (3x² + 3x - 5)
-2x² + 4x - 2 + 3x² + 3x - 5
x² + 7x - 7

5. Which sum is irrational?

(1) -2√(12) + √(100)
(2) -√(4) + 1/3 √(900)
(3) 1/2 √(25) + √(64)
(4) √(49) + 3√(121)

Answer: (1) -2√(12) + √(100)

You could put each of these into your calculator and see if you get an infinite, non-repeating decimal as answer. If the numbers under the radicals are perfect squares, then the numbers are rational numbers. The sum of two rational numbers is always rational. The sum of a rational and an irrational number is also irrational. (The sum of two irrational numbers is usually, but not always, irrational.)

Looking at the choices above, 12 is not a perfect square, but 100, 4, 900, 25, 64, 49, and 121 are perfect squares.

Choice (1) is the only choice that adds an irrational number to a rational number and is therefore irrational. This is the correct answer.

Choice (2) equals -2 + 10 = 8.

Choice (3) equals 2.5 + 8 = 10.5

Choice (4) equals 7 + 33 = 40.

6. The solution to (4(x - 5)) / 3 + 2 = 14 is

(1) 15
(2) 14
(3) 6
(4) 4

Answer: (2) 14

You can substitute each answer in your calcluator if you aren't sure how to solve this.

Use inverse operations to isolate the variable and solve for x:

(4(x - 5)) / 3 + 2 = 14
(4(x - 5)) / 3 = 12
4(x - 5) = 36
x - 5 = 9
x = 14

Check: (4(14 - 5)) / 3 + 2 = (4(9)) / 3 + 2 = (36) / 3 + 2 = 12 + 2 = 14. Check!

The correct answer Choice (2).

7. On an island, a rare breed of rabbit doubled its population each month for two years. Which type of function best models the increase in population at the end of two years?

(1) linear growth
(2) linear decay
(3) exponential growth
(4) exponential decay

Answer: (3) exponential growth

Continously doubling is growth not decay. The rate of change is not constant, so it is not linear.

This is an example of exponential growth. This is a definition question. There is nothing to solve. You either understand the concept or you do not.

8. What is the degree of the polynomial 2x - x² + 4x³?

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

Answer: (3) 3

This is another concept/definition question. The degree of a polynomial is the value of the highest exponent. This is the term that would go first when written in standard form. The correct answer is 3, which is Choice (3).

Choice (1), 1, is the value of the exponent of the first term that is written, but it isn't the highest exponent. Eliminate Choice (1).

Choice (2), 2, is the coefficient of the first term that is written, not the exponent. It's the exponent of the second term, but that doesn't matter. Eliminate Choice (2).

Choice (4), 4, is the leading coefficient, meaning that it would be the coefficient of the first term if the expression was written in the correct order (standard form). Eliminate Choice (4)

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

June 2024 Algebra Regents Part IV

This exam was adminstered in January 2024 .

June 2024 Algebra, Part IV

Each correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Courtney went to a coffee shop to purchase lattes and donuts for her friends. One day she spent a total of $15.50 on four lattes and two donuts. The next day she spent a total of $18.10 on three lattes and five donuts. All prices included tax.

If x represents the cost of one latte and y represents the cost of one donut, write a system of equations that can be used to model this situation.

Courtney thinks that one latte costs $2.75 and one donut costs $2.25.
Is Courtney correct? Justify your answer.

Use your equations to determine algebraically the exact cost of one latte and the exact cost of one donut.

Answer:

Answer every section of this question to get full credit. if you make a mistake on top, you will not be penalized more than once if you are consistent until the end and give a final answer. Make sure you use x and y and not L and D because the questions says to use x and y!

You should be able to read the first two sentences and translate them into equations.

4x + 2y = 15.50
3x + 5y = 18.10

Two find out if Courtney is correct, you only have to substitute those values into BOTH equations to see if those values make BOTH equations true.

4(2.75) + 2(2.25) = 15.50 (check)
3(2.75) + 5(2.25) = 19.50 =/= 18.10 (does not check!)

Courtney is incorrect.

Finally, copy your system of equations and solve through elimination. Note that it says algebraically, so you won't get full credit for graphing the equations in your calculator and finding the intersection point.

4x + 2y = 15.50
3x + 5y = 18.10

12x + 6y = 46.50
12x + 20y = 72.40

14y = 25.90

y = $1.85

The price of one donut is $1.85. If you got an repeating decimal, or anything that doesn't come out to an exact number of dollars and cents, that should tell you that there was an error somewhere. The problem deals with money and must have a monetary answer.

Now that you know the price of a donut, you can find the price of a latte by using either of the original equations:

4x + 2y = 15.50
4x + 2(1.85) = 15.50
4x + 3.70 = 15.50
4x = 11.80
x = 2.95.

The price of a latte is $2.95.

End of exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.

June 2024 Algebra Regents Part III

This exam was adminstered in January 2024 .

June 2024 Algebra, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

31. Graph the following system of equations on the set of axes below.

y = x² - 3x - 6
y = x - 1
State the coordinates of all solutions.

Answer:

Use your calculator and check the table of values to find the points. Only label the points of intersection, and be sure to label at least one line.

Your graph should look like this one:

The solutions are the points of intersection: (-1,-2) and (5,4)

32. The table below shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.

Write the linear regression equation for this data set, rounding all values to the nearest hundredth.

State the correlation coefficient to the nearest hundredth.

State what this correlation coefficient indicates about the linear fit of the data.

Answer:

For a linear regression, put all of the data into two lists on your graphing calculator and run the regression. Make sure you have "DiagonsticOn" in the Catalog.

The equation will be y = -37.57x + 215.67

The correlation coefficient is -.98.

This means that there is a strong negative correlation between the weeks and the dollars earned.

33. Use the quadratic formula to solve the equation 3x² - 10x + 5 = 0. Express the answer in simplest radical form.

Answer:

You must use the Quadratic Formula, which will likely be easier than completing the square. Factoring won't work because you know the answer will have a radical in it.

The Quadratic Formula is x = (-b + √(b² - 4ac) ) / (2a), where a, b and c are the coefficients of the terms.

x = (-(-10) + √((-10)² - 4(3)(5)) ) / (2(3))
x = (10 + √(100 - 60) ) / (6)
x = (10 + √(40) ) / (6)
x = (10 + 2 √(10) ) / (6)
x = 5/3 + 1/3 √(10)

34. Graph the system of inequalities on the set of axes below.

3y + 2x < 15
y - x > 1

State the coordinates of a point in the solution to this system. Justify your answer.

Answer:

Rewrite the inequalities into slope-intercept form and put them into your calculator. Check the table of values for points on the line.

The first inequality will have a solid line. The second will have a broken or dashed line. Points on the broken line are NOT in the solution set.

You're going to graph below the first line and above the second line.

3y + 2x < 15
3y < -2x + 15
y < -2/3x + 5

y - x > 1
y > x + 1

You're graph should look like this:

Pick any point in the double-shaded area that is NOT on the broken line.

For example, (-10,0) is in the solution set because it's in the double-shaded area where the big S is labeled.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and 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.