## Wednesday, April 25, 2018

### Algebra 2 Problems of the Day (open-ended)

Continuing with daily Algebra 2 questions and answers.

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January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.

31.The zeros of a quartic polynomial function h are 1, +2, and 3.
Sketch a graph of y = h(x) on the grid below.

They are asking for a "sketch", not an exact graph, because they did not give you the actual function. There are an infinite number of quartic (fourth power) graphs that have these zeroes.
You can enter the following into your graphing calculator to see an example:

y = (x + 2)(x + 1)(x - 2)(x - 3)

You can also multiply that by any coefficient, positive or negative. As always, a positive multiplier means that the graph opens upward, and a negative means that it will open downward. Either is acceptable.

Note that you needed to have the zeroes in the correct places for full credit. If you flipped the signs, for example, you would lose a credit.

32. 2 Explain why 813/4 equals 27.

The fourth root of 81 is 3 and 3 to the third power is 27.
You can write it as 81 = (3 * 3 * 3 * 3)3/4 = (3 * 3 * 3) = 27 and add an explanation.

Reminder "Explain" means that you have to write words to "explain" (sorry about the repetition, except that I'm not). "Justify" means that you can just write equations as proof, but "explain" means at least a sentence.

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## Tuesday, April 24, 2018

### Algebra 2 Problems of the Day (open-ended)

Continuing with daily Algebra 2 questions and answers.

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January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.

29. Researchers in a local area found that the population of rabbits with an initial population of 20 grew continuously at the rate of 5% per month. The fox population had an initial value of 30 and grew continuously at the rate of 3% per month.
Find, to the nearest tenth of a month, how long it takes for these populations to be equal.

The equation for rabbit population is y = 20e.05x.
The equation for fox population is y = 30e.03x.
Set them equal to each other 20e.05x = 30e.03x
Divide by 30: (20/30)e.05x = e.03x
Divide by the e term: (2/3) = (e.03x)/(e.05x)
Simplify the exponent: 2/3 = e-.02x
ln (2/3) / -.02 = -.02x / -.02
20.273... = x
20.3 months.

You also could have graphed the two equations, found the point of intersection, record that point of intersection on your exam paper as an ordered pair, and then state the answer of 20.3 months based on the x coordinate.

30. Consider the function h(x) = 2sin(3x) + 1 and the function q represented in the table below.

Determine which function has the smaller minimum value for the domain [-2,2]. Justify your answer.

The minimum for q(x) is -8, according to the table. You don't need to work out what the equation is.
The minimum for h(x) is -1. You don't need to explain how you know this as it would likely be assumed that you got it from the calculator, or you just knew that y = 2 sin(3x) has a minimum of -2 and a maximum of 2, so if you add 1 to those numbers, the minimum is -1.
So q(x) has the smaller minimum. Don't forget to mention this. It isn't enough to just state the two minimums. But you DO need to state the two minimums.

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## Monday, April 23, 2018

### Boyfriend

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

Because it is a high school, and these things happen, even when I can't work math into it.

Come back often for more funny math and geeky comics.

### Algebra 2 Problems of the Day (open-ended)

Continuing with daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.

27.A formula for work problems involving two people is shown below.

1/t1 + 1/t2 = 1/tb
t1 = the time taken by the first person to complete the job
t2 = the time taken by the second person to complete the job
tb = the time it takes for them working together to complete the job

Fred and Barney are carpenters who build the same model desk. It takes Fred eight hours to build the desk while it only takes Barney six hours. Write an equation that can be used to find the time it would take both carpenters working together to build a desk.
Determine, to the nearest tenth of an hour, how long it would take Fred and Barney working together to build a desk.

The equation needed to solve this would be

1/8 + 1/6 = 1/tb or 1/8 + 1/6 = 1/x

Once you have this, you can solve for x (or tb).

3/24 + 4/26 = 1/x
7/24 = 1/x
x = 24/7 = 3.428571... = 3.4 hours

Alternatively, multiply the entire equation by (8)(6)(x) to eliminate the fractions:

(8)(6)(x)(1/8) + (8)(6)(x)(1/6) = (8)(6)(x)(1/x)
6x + 8x = 48
14x = 48
x = 48 / 14 = 3.4 hours.

Also of note, whoever wrote this questions obviously likes The Flintstones.

28.Completely factor the following expression:

x2 + 3xy + 3x3 + y

First, rewrite the expression as 3x3 + x2 + 3xy + y
Factor by grouping: x2(3x + 1) + y(3x + 1)
Factor again: (x2 + y)(3x + 1)

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## Sunday, April 22, 2018

### Algebra 2 Problems of the Day (open-ended)

Continuing with daily Algebra 2 questions and answers.

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January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.
Elizabeth tried to find the product of (2 + 4i) and (3 - i), and her work is shown below.

(2 + 4i)(3 - i)
= 6 - 2i + 12i - 4i2
= 6 + 10i - 4i2
= 6 + 10i - 4(1)
= 6 + 10i - 4
= 2 + 10i

Identify the error in the process shown and determine the correct product of (2 + 4i) and (3 - i).

Elizabeth replaced i2 with 1 instead of -1.

= 6 + 10i - 4(-1)
= 6 + 10i + 4
= 10 + 10i

26.A runner is using a nine-week training app to prepare for a “fun run.” The table below represents the amount of the program completed, A, and the distance covered in a session, D, in miles.

Based on these data, write an exponential regression equation, rounded to the nearest thousandth, to model the distance the runner is able to complete in a session as she continues through the nine-week program.

Enter the data into two lists (L1 and L2, most likely). Check for errors.
Go to STAT, CALC and select ExpReg.
You should get the following output:
y = a*b^x
a = 1.223034549
b = 2.652024589
Round these numbers to the nearest thousandth. (You will lose a point if you do not round correctly.)
y = 1.223(2.652)x.
You could have used A for x and D for y in your answer.

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## Saturday, April 21, 2018

### Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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January 2018
23. If the function g(x) = acx represents exponential growth, which statement about g(x) is false?

(1) a > 0 and b > 1
(2) The y-intercept is (0, a).
(3) The asymptote is y = 0.
(4) The x-intercept is (b, 0).

Answer: (4) The x-intercept is (b, 0).
The function has no x-intercept. And when x = b, then g(x) = abb, not 0.
Note that choice (3) and (4) are mutually exclusive, so one of them has to be false.
Since it is exponential growth, a > 0 and b > 1. And when x = 0, g(x) = ab0 = a.

24.At her job, Pat earns \$25,000 the first year and receives a raise of \$1000 each year. The explicit formula for the nth term of this sequence is an = 25,000 + (n - 1)1000. Which rule best represents the equivalent recursive formula?
(1) an = 24,000 + 1000n
(2) an = 25,000 + 1000n
(3) a1 = 25,000, an - 1 + 1000
(4) a1 = 25,000, an + 1 + 1000

Answer: (3) a1 = 25,000, an - 1 + 1000
In the recursive formula, each term is the sum of the term before it (an-1) plus 1000, which is choice (3).
Note that choices (1) and (2) are not recursive formulas.

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## Friday, April 20, 2018

### Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2018
21. What is the inverse of f(x) = -6(x - 2)?

(1) f-1(x) = -2 - x/6
(2) f-1(x) = 2 - x/6
(3) f-1(x) = 1 / (-6(x - 2))
(4) f-1(x) = 6(x - 2)

Answer: (2) f-1(x) = 2 - x/6
Inverse operations. Divide by negative six, then add two.
x = -6(f-1(x) - 2)
x / (-6) = f-1(x) - 2
2 - x/6 = f-1(x).

22. Brian deposited 1 cent into an empty non-interest bearing bank account on the first day of the month. He then additionally deposited 3 cents on the second day, 9 cents on the third day, and 27 cents on the fourth day. What would be the total amount of money in the account at the end of the 20th day if the pattern continued?
(1) \$11,622,614.67
(2) \$17,433,922.00
(3) \$116,226,146.80
(4) \$1,743,392,200.00

Do not answer the 20th term in the geometric sequence. They are looking for the sum of the first 20 terms.
The formula for finding the sum of the first n terms in a geometric sequence is

Sn = (a1(1 - rn)) / (1 - r),

where n is the number of terms, r is the common ratio, and a1 is the initial term.
In this question, the common ratio is 3, because the sequence goes 1, 3, 9, 27 ...
So the sum is (1 * (1 - 320)/(1 - 3) = 1743392200, which is the number of cents. Divide this by 100 to convert it to dollars, or \$17,433,922.00.
Alternatively, you could have used .01 for the initial term in the formula, which would have given you the answer immediately.

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## Thursday, April 19, 2018

### Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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January 2018
19. If p(x) = 2x3 - 3x + 5, what is the remainder of p(x) : (x - 5)?

(1) -230
(2) 0
(3) 30
(4) 240

The Polynomial Remainder Theorem tells us that is p(x) is divided by (x - r), then the remainder, R, can be found by evaluating p(r).
If (x - 5) is a factor of p(x), then when x = 5, p(x) would = 0. If it is not a factor, then the value of p(5) will be the remainder when you divide the polynomials.
If you calculate p(5), you will get 2(5)3 - 3(5) + 5 = 240, which is the remainder.
Alternatively, if you forgot this, you can do the polynomial division. This will give you 240 as a remainder. See the image below:

20. The results of simulating tossing a coin 10 times, recording the number of heads, and repeating this 50 times are shown in the graph below.

Based on the results of the simulation, which statement is false?
(1) Five heads occurred most often, which is consistent with the theoretical probability of obtaining a heads.
(2) Eight heads is unusual, as it falls outside the middle 95% of the data.
(3) Obtaining three heads or fewer occurred 28% of the time.
(4) Seven heads is not unusual, as it falls within the middle 95% of the data.

Answer: (2) Eight heads is unusual, as it falls outside the middle 95% of the data.
Eight does not fall outside the middle 95% of the data. There are 50 data points, so 47.5 pieces of data are in the middle, leaving 2.5 / 2 = 1.25 pieces of data more than two standard deviations above and below the mean. But there are two results greater than 8, so it's not outside of the middle 95%.

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## Wednesday, April 18, 2018

### Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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January 2018
17. The function below models the average price of gas in a small town computations. since January 1st.
G(t) = -0.0049t4 + 0.0923t3 - 0.56t2 + 1.166t + 3.23, where 0 ≤ t ≤ 10.

If G(t) is the average price of gas in dollars and t represents the number of months since January 1st, the absolute maximum G(t) reaches over the given domain is about

(1) \$1.60
(2) \$3.92
(3) \$4.01
(4) \$7.73

Graph the function and use "maximum" to find the highest value, which you should see is just above \$4.00.
See the graph below:

At approximately t = 1.6, G(t) = 4.01, approximately.

18. Written in simplest form, (c2 - d2) / (d2 + cd - 2c2), where c =/= d, is equivalent to
(1) (c + d) / (d + 2c)
(2) (c - d) / (d + 2c)
(3) (-c - d) / (d + 2c)
(4) (-c + d) / (d + 2c)

Answer: (3) (-c - d) / (d + 2c)
The numerator, (c2 - d2), is the difference of two perfect squares, and factors into the conjugates, (c + d)(c - d).
Note that all four choices have (d + 2c) as the denominator, which makes factoring (d2 + cd - 2c2) that much easier into (d + 2c)(d - c).
(c - d) / (d - c) = -1, which reduces the fraction to (-1)(c + d) / (d + 2c).
Distribute the -1, and you get choice (3).

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## Tuesday, April 17, 2018

### E-mote

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

They're irrational, you know.

I remember when they were just ''smileys''. Then ''emoticons'' (emote icons). Finally, ''emoji''. Like ''Gojira'' instead of ''Godzilla''.

Come back often for more funny math and geeky comics.

### Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

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January 2018
15. The terminal side of Î¸, an angle in standard position, intersects the unit circle at P(-1/3, -sqrt(8)/3). What is the value of sec Î¸?

(1) -3
(2) -3*sqrt(8)/8
(3) -1/3
(4) -sqrt(8)/3

The coordinates of P are (cos Î¸, sin Î¸)
sec Î¸ = 1 / cos Î¸
cos Î¸ = -1/3
sec Î¸ = 1 / (-1/3) = -3

16. What is the equation of the directrix for the parabola -8(y - 3) = (x + 4)2?

(1) y = 5
(2) y = 1
(3) y = -2
(4) y = -6

When the parabola is written in this form -- (x − p)2=±4a(y−q) -- then (p,q) will be the vertex and a is the focus length. In other words, the distance in one direction from the vertex will be the focus, and in the other direction will be the directrix.

The vertex is (-4, 3) and the focal length is 2. The negative tells us that the parabola is opening down, so the directrix is 2 units above the vertex, which is y = 5.

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## Monday, April 16, 2018

### Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
13. If aebt = c, where a, b, and c are positive, then t equals

Divide both sides by a: ebt = c/a
Take the natural log: ln(ebt) = ln(c/a)
which gives you: bt = ln(c/a)
Divide by b: t = ln(c/a) / b.

14. For which values of x, rounded to the nearest hundredth, will |x2 - 9| - 3 = log3x?

(1) 2.29 and 3.63
(2) 2.37 and 3.54
(3) 2.84 and 3.17
(4) 2.92 and 3.06