Friday, May 18, 2018

Algebra 2 Problems of the Day (open ended)

Daily Algebra 2 questions and answers.

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August 2017, Part II

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


31. Algebraically determine whether the function j(x)= x4 - 3x2 - 4 is odd, even, or neither.

Answer:
A function is even if it is symmetrical about the y-axis. If reflected across the y-axis, it will be mapped onto itself. Because of this symmetry, for any value of x, f(x) = f(-x).
A function is odd if it is rotational symmetrical about the origin. If rotated 180 degrees about the origin, it will be mapped onto itself. Because of this symmetry, for any value of x, f(-x) = -f(x).

All the exponents (including the constant) have even exponents, so we know that the function is even. We need to show it algebraically, but knowing this tells you which situation you should start with.

f(-x) = (-x)4 - 3(-x)2 - 4
f(-x) = x4 - 3x2 - 4
f(-x) = f(x), therefore f(x) is even.





32. On the axes below, sketch a possible function p(x) = (x - a)(x - b)(x + c), where a, b, and c are positive, a > b, and p(x) has a positive y-intercept of d. Label all intercepts.


Answer:
In the given function, a and b are roots on the right side of the y-axis (positive), and -c will be on left side (negative). Also, b comes before a because a > b. Finally, d is on the y-axis somewhere above the x-axis. This is just a sketch, it doesn't have to be perfect. At a minimum, please make sure your sketch passes the vertical line test -- don't be sloppy.

The line has to go from -c to d, so it starts in Quadrant III, through -c to d, then down to b and back up through a and beyond.
See the image below.
Note: d does not have to be a local maximum, but it could be. And c is a positive number, so the axis must be labeled -c.





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Thursday, May 17, 2018

Algebra 2 Problems of the Day (open ended)

Daily Algebra 2 questions and answers.

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August 2017, Part II

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


29. While experimenting with her calculator, Candy creates the sequence 4, 9, 19, 39, 79, … .
Write a recursive formula for Candy’s sequence.
Determine the eighth term in Candy’s sequence.

Answer:
If you look at the differences, you see that the sequence increases by 5, then 10, then 20, then 40, etc. It does not have a common difference, but it does have a common ratio of 2, so it's a geometric sequence.
The initial value is 4, and 4 times 2 is 8, so you have to add another 1 to get 9, the second number in the sequence.

The recursive formula for this sequence will be:

a1 = 4
an = an-1 + 1

To get the 8th term in the sequence, just continue where the question left off. You were given the first 5 terms.
a6 = a5 + 1 = 2(79) + 1 = 159
a7 = a6 + 1 = 2(159) + 1 = 319
a8 = a7 + 1 = 2(319) + 1 = 639





30. In New York State, the minimum wage has grown exponentially. In 1966, the minimum wage was $1.25 an hour and in 2015, it was $8.75. Algebraically determine the rate of growth to the nearest percent.

Answer:
2015 - 1966 = 49 years, which will be the exponent.
8.75 = 1.25(x)49
7 = x49
(7)(1/49) = x --> Yes, you are going to take the 49th root of 7.
x = 1.04051.., which is approximately 104%
This is about 4% growth to the nearest percent.

(Don't stop at 1.04, or round that to 1)



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Octillion

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(C)Copyright 2018, C. Burke.

I skipped the 'what to wear' joke -- you figure she'll round up something.




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Tuesday, May 15, 2018

Algebra 2 Problems of the Day (open ended)

Daily Algebra 2 questions and answers.

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August 2017, Part II

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


27. Verify the following Pythagorean identity for all values of x and y: (x2 + y2)2 = (x2 − y2)22 + (2xy)2

Answer:


(x2 + y2)2 = (x2 − y2)22 + (2xy)2
x4 + 2x2y2 + y4 = x4 - 2x2y2 + y4 + 4x2y2
x4 + 2x2y2 + y4 = x4 + 2x2y2 + y4
(check mark!)

As long as you showed the left side and right side to be equal, you should be fine.
You could have only done the right side, leaving the left side as (x2 + y2)2, and as a final step, reduce the right side to (x2 + y2)2 as well.





28. Mrs. Jones had hundreds of jelly beans in a bag that contained equal numbers of six different flavors. Her student randomly selected four jelly beans and they were all black licorice. Her student complained and said "What are the odds I got all of that kind?" Mrs. Jones replied, "simulate rolling a die 250 times and tell me if four black licorice jelly beans is unusual." Explain how this simulation could be used to solve the problem.

Answer:
A die has six sides, each of which is just as likely to occur. Each of the six sides could represent a color of a jelly bean in the bag. If, for example, 1, represents "black licorice", you could use the simulation to see how many times that this number would be rolled 4 times in a row after 250 simulations.

Make sure that you explain the simulation for full credit.



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The Refrigerator

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(C)Copyright 2018, C. Burke.

Those fridge magnets have to hold up something. Just wait until you learn about magnets in Science!

That's Hank in the doorway. The two young ladies are new. Names pending. (Suggestions?)




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Monday, May 14, 2018

Algebra 2 Problems of the Day (open-ended)

Daily Algebra 2 questions and answers.

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August 2017, Part II

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


25.Explain how (−8)(4/3) can be evaluated using properties of rational exponents to result in an integer answer

Answer:
A rational exponent, like (4/3), can be re-written as (4) * (1/3), which means to take the fourth power of the cube root of the expression (-8). (The power of a power rule)
The cube root of -8 is -2, which when raised to the 4th power is (-2)4 = 16, which is an integer.





26. A study was designed to test the effectiveness of a new drug. Half of the volunteers received the drug. The other half received a sugar pill. The probability of a volunteer receiving the drug and getting well was 40%. What is the probability of a volunteer getting well, given that the volunteer received the drug?

Answer:
The probability of two things occurring is 40%. The combined probability can be found by multiplying the two independent probabilities.
Since we know one of those probabilities, we can solve for the other.
P(A) * P(B) = .40
1/2 * P(B) = .40
P(B) = .80
There was an 80% chance that if the volunteer received the drug that he (or she) would get better.



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Sunday, May 13, 2018

Happy Mothers Day 2018

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(C)Copyright 2018, C. Burke.

Worry is a function of motherhood.

As for those transformations, if the parent function f(x) = x2 is transformed into g(x), then g(x) = -(x + 11)2 + 3

If the parent function h(x) = |x| is transformed into j(x), then j(x) = 2|x - 14)2 + 2

(I realize that I haven't been doing Mothers Day for a while. I hope you can understand.)




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Thursday, May 10, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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August 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


22. A public opinion poll was conducted on behalf of Mayor Ortega's reelection campaign shortly before the election. 264 out of 550 likely voters said they would vote for Mayor Ortega; the rest said they would vote for his opponent.
Which statement is least appropriate to make, according to the results of the poll?

1) There is a 48% chance that Mayor Ortega will win the election.
2) The point estimate (ˆp) of voters who will vote for Mayor Ortega is 48%.
3) It is most likely that between 44% and 52% of voters will vote for Mayor Ortega.
4) Due to the margin of error, an inference cannot be made regarding whether Mayor Ortega or his opponent is most likely to win the election.

Answer: There is a 48% chance that Mayor Ortega will win the election.
Analysis of data. While 48% of the likely voters in this poll said that they would vote for Mayor Ortega, this is only a sample of all the voters and there is a margin of error. It is more appropriate to say that the point estimate is 48% than to say that this is his chance of winning.





23. What does ( (-54x9) / y4 )2/3 equal?


Answer:4)
Check your work against the image below.
We need to take the cube root of the expression. We can factor -54 into (-27)(2) because -27 is a perfect cube. The cube root of x9 is x3. The y4 term can be broken into y3 times y, and the cube root of y3 is y.
After that we can square all the terms. (You could have done that sooner, if you don't mind squaring -54 and then finding the largest perfect cube that's a factor.)
(-3x3) squared is 9x6, over y2, times the cube root of 4 over y2. Split up the radical into a fraction of two radicals and you have choice 4.





The Rickerts decided to set up an account for their daughter to pay for her college education. The day their daughter was born, they deposited $1000 in an account that pays 1.8% compounded annually. Beginning with her first birthday, they deposit an additional $750 into the account on each of her birthdays. Which expression correctly represents the amount of money in the account n years after their daughter was born?

1) an = 1000(1.018)n + 750
2) an = 1000(1.018)n + 750n
3) a0 = 1000; an = an - 1 (1.018) + 750
4) a0 = 1000; an = an - 1 (1.018) + 750n

Answer:3) a0 = 1000; an = an - 1 (1.018) + 750
The initial value was $1000. Each year the value of the account will by 1.8% greater plus another $750 will be added to the balance. So the previous value, an - 1, times (1.018) then add 750.

Choice 1 doesn't work because the $750 is only added once, instead of every year, and it isn't earning any interest.
Choice 2 doesn't work because the $750, while being added each year, isn't earning the 1.8% interest.
Choice 4 shows a recursive formula, so no variable is needed at $750. The parents aren't adding, for example, $1500 on her second birthday or $3000 on her fourth.



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Wednesday, May 09, 2018

Sinusoidal

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(C)Copyright 2018, C. Burke.

Well, sure! We went straight from cold season to allergy season this year!




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Tuesday, May 08, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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August 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


19. Consider the system shown below.

2x − y = 4
(x + 3)2 + y2 = 8

The two solutions of the system can be described as

1) both imaginary
2) both irrational
3) both rational
4) one rational and one irrational

Answer: 1) both imaginary
Rewrite the first equation as y = 2x - 4, then substitute it into the second equation:

(x + 3)2 + (2x - 4)2 = 8
x2 + 6x + 9 + 4x2 - 16x + 16 = 8
5x2 - 10x + 25 = 8
5x2 - 10x + 17 = 0

a = 5, b = -10, c = 17
Discriminant: b2 - 4ac = (-10)2 - 4(5)(17) = -240, which is less than 0.
Roots are imaginary.

Had the discriminant been positive, you could have continued using the Quadratic Formula to determine if the roots were both rational or both irrational. One of each wouldn't happen.





20. Which binomial is not a factor of the expression x3 − 11x2 + 16x + 84?

1) x + 2
2) x + 4
3) x - 6
4) x - 7

Answer:2) x + 4
A quick graph of x3 − 11x2 + 16x + 84 shows zeroes at -2, 6, and 7, meaning that (x + 2), (x - 6), and (x - 7) are factors.

If you want to work it out using division, you will see that (x + 4) leaves a remainder, so it is not a factor of the polynomial. The other three do not leave remainders. (see image below)





21. A ball is dropped from a height of 32 feet. It bounces and rebounds 80% of the height from which it was falling. What is the total downward distance, in feet, the ball traveled up to the 12th bounce?

1) 29
2) 58
3) 120
4) 149

Answer:4) 149
If it falls from 32 feet, how could the distance be 29 feet? Twelve bounces later, 58 is "obviously" incorrect as well. Eliminate those two before you do anything else.

The total distance up until that the nth bounce is
Sn = (32 - 32(.8)12) / (1 - .8) = 149.004883722... = 149



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Sunday, May 06, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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August 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


16. A study conducted in 2004 in New York City found that 212 out of 1334 participants had hypertension. Kim ran a simulation of 100 studies based on these data. The output of the simulation is shown in the diagram below.


At a 95% confidence level, the proportion of New York City residents with hypertension and the margin of error are closest to

1) proportion ≈ .16; margin of error ≈.01
2) proportion ≈ .16; margin of error ≈.02
3) proportion ≈ .01; margin of error ≈.16
4) proportion ≈ .02; margin of error ≈.16

Answer: 2) proportion ≈ .16; margin of error ≈.02
Hopefully, it's "obvious" from the choices that the proportion is approximately .16, not .01 or .02. And the margin of error is not .16.
The question is, which is the margin of error, .01 or .02?

You can find the margin of error for proportions using the formula
ME = z( sqrt( ((p)(1-p))/n ) ),
where p is the proportion, n is the number of participants and z is the z-score, which 1.96 for 95% confidence.

So ME = (1.96)( sqrt( ((0.16)(0.84))/1334 ) ) = 0.019673... or about 0.02.





17. Which scenario is best described as an observational study?

1) For a class project, students in Health class ask every tenth student entering the school if they eat breakfast in the morning.
2) A social researcher wants to learn whether or not there is a link between attendance and grades. She gathers data from 15 school districts.
3) A researcher wants to learn whether or not there is a link between children's daily amount of physical activity and their overall energy level. During lunch at the local high school, she distributed a short questionnaire to students in the cafeteria
4) Sixty seniors taking a course in Advanced Algebra Concepts are randomly divided into two classes. One class uses a graphing calculator all the time, and the other class never uses graphing calculators. A guidance counselor wants to determine whether there is a link between graphing calculator use and students' final exam grades.

Answer: 2) A social researcher wants to learn whether or not there is a link between attendance and grades. She gathers data from 15 school districts.
An observational study draws inferences from a sample where the independent variable is not under the control of the researcher. In the second scenario, the researcher is removed from the subjects of the study. In choices 1 and 3, there is interaction between the researcher and the subjects. In choice 4, preventing students from using calculators to see if it affects their final exam grades wouldn't be an ethical study.





18. Which sinusoid has the greatest amplitude?


Answer:4) y = −5sin(θ − 1) − 3
You can see from the graphs that Choice 3 has a greater amplitude than Choice 1, which we can eliminate.
You can see from the values in front of the equations, 3 and -5, that Choice 4 has a greater amplitude than Choice 2, which we can eliminate.
Choice 4 has an Amplitude of 5. Choice 2 has an Amplitude of 8/2 = 4. (You need to divide it by 2 because the amplitude is measured from the center line, not top to bottom.)



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Saturday, May 05, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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August 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


13. Which expression is equivalent to (4x3 + 9x − 5) / (2x − 1) , where x ≠ 1/2 ?
Which equation approximates the amount of Iridium-192 present after t days?

1) 2x2 + x + 5
2) 2x2 - x + 5
3) 2x2 + 11/2 + 1 / (2(2x-1))
4) 2x2 - x + 4 + 1 / (2x - 1)

Answer: 1) 2x2 + x + 5
First of all, "x ≠ 1/2" is specified because the expression would be undefined because the denominator of the fraction would be zero. It isn't needed for anything else.

You could multiply choices 1 and 3 by 2x - 1 to see if you got 4x3 + 9x − 5. The other two choices would be messier, but you would have your answer before you got to them.

Or you can do the polynomial division:





14. The inverse of the function f(x) = (x + 1) / (x − 2) is

1) f-1 = (x + 1)/(x + 2)
2) f-1 = (2x + 1)/(x - 1)
3) f-1 = (x + 1)/(x - 2)
4) f-1 = (x - 1)/(x + 1)

Answer: 2) f-1 = (2x + 1)/(x - 1)
Write the function at y = (x + 1) / (x - 2)
Switch the x's and y's: x = (y + 1) / (y - 2)
Solve for y by getting all the y variables on the left side.
Any term without a y in it goes to the right side.
Finally, factor the y from the expression on the left, and then divide.

x (y - 2) = y + 1
xy - 2x = y + 1
xy - y - 2x = 1
xy - y = 2x + 1
y(x - 1) = 2x + 1
y = (2x + 1)(x - 1)





15. Which expression has been rewritten correctly to form a true statement?


1) (x + 2)2 + 2(x + 2) − 8 = (x + 6)x
2) x4 + 4x2 + 9x2y2 − 36y2 = (x + 3y)2(x − 2)2
3) x3 + 3x2 − 4xy2 − 12y2 = (x - 2y)(x + 3)2
4) (x2 − 4)2 − 5(x2 − 4) − 6 = (x2 - 7)(x2 - 6)

Answer:4) B and D
You can multiply these, and in this case, it won't take so long because the first one is correct. Or you can put each side of each equation into the graphing calculator and compare the Tables of Values.

You can also simplify the multiplication by using substitutions.
Suppose you let y = (x + 2), then choice 1 becomes y2 + 2y - 8 = (y + 4)(y - 2)
If you factor the polynomial, the factors of -8 that add to +2 are +4 and -2, so this works.

You could have done something for choice 4, letting w = x2 - 4, so that the equation could be be rewritten as:

w2 - 5w - 6 = (w - 3)(w - 2)

However, if you multiply those binomials, you'll get + 6, not - 6.

You can see that in Choice 2, the left side of the equation ends with - 36y2. However, on the right side, 3y2 is 9y2 and (-2)2 is 4, both positive numbers, which is a contradiction.
In Choice 3, you can see on the right side that the y term is not squared, but y2 appears in two terms. This isn't possible.



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