(C)Copyright 2017, C. Burke.
With apologies to Debussy.
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With apologies to Debussy.
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Before he was president, he wanted to be a mathematician. He didn't get to be an ex-president.
Happy Presidents Day!
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He didn't say whether or not it was a *good* point.
UPDATE: When I asked the question in class, one young lady said, "What's the point?"
I think I like that better!
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That's the right formula, right? Or h could be happiness.
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Pretty much gave up the pun in the title this time, huh?
You know, "2b | ~2b" works as well, but it doesn't read the same!
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And she didn't try to dissuade him at all! What a friend!
But Judy was there on that fateful day back in September, as seen here.
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This question was worth 6 credits
1.Which expression is equivalent to 16x^{2} - 36?
(2) 4(2x + 3)(2x - 3). This question was a giveaway. It the Difference of Squares which can be factored into Conjugates, that is, the same binomial except that one has a plus and the other a minus. Only one choice has that!
Additionally, if you forgot how to factor, you could just multiply the choices and see which gives you the original expression.
You could factor this by dividing by 4, getting 4(4x^{2} - 9) which becomes 4(2x + 3)(2x - 3).
Or you could have factored it into (4x + 6)(4x - 6), each of which could have a factor of 2 taken out of them.
2.What is the solution set of the equation (x - 2)(x - a) = 0?
(3) 2 and a. Flip the signs: 2 - 2 = 0 or a - a = 0. Don't let the a throw you off.
3.Analysis of data from a statistical study shows a linear relationship in the data with a correlation coefficient of -0.524. Which statement best summarizes this result?
(4) There is a moderate negative correlation between the variables. Hopefully, you immediately eliminated the "positive" choices. A coefficient of -0.5 (or +0.5) can best be described as "moderate", as opposed to "strong" or "weak".
4.Boyle's Law involves the pressure and volume of gas in a container. It can be represented by the formula
P_{1}V_{1} = P_{2}V_{2}. When the formula is solved for P_{2}, the result is
(3) P_{1}V_{1} / V_{2}. Divide both sides by V2 to isolate P2
5.A radio station did a survey to determine what kind of music to play by taking a sample of middle school, high school, and college students. They were asked which of three different types of music they prefer on the radio: hip-hop, alternative, or classic rock The results are summarized in the table below.
Hip-Hop | Alternative | Classic Rock | |
Middle School | 28 | 18 | 4 |
High School | 22 | 22 | 6 |
College | 16 | 20 | 14 |
(2) 28%. The first two rows are irrelevant. Only the college row is important. There were a total of 50 college students surveyed and only 14 prefer classic rock. That's 28%.
6.Which function has zeros of -4 and 2?
(4) The graph shows the function crossing the x-axis at -4 and 2. Not a trick question.
The equation for the function would involve multiplying the factors (x + 4)(x - 2), which would have a middle term of 2x, not 7x nor -7x.
Likewise, if the zeros are -4 and 2, then the Axis of Symmetry must be -1, which is exactly in the middle. If you use the formula
x = (-b/2a), it is obvious that b cannot be either 7 or -7.
7.Which expression is equivalent to 2(3g - 4) - (8g + 3)?
(4) -2g - 11.
8.In 2014, the cost to mail a letter was 49¢ for up to one ounce. Every additional ounce cost 21¢. Which recursive function could be used to determine the cost of a 3-ounce letter, in cents?
(1) a_{1} = 49; a_{n} = a_{n-1} + 21. The first ounce is 49 cents, and each additional is 21.
9.A car leaves Albany, NY, and travels west toward Buffalo, NY. The equation D = 280 - 59t can be used to represent the distance, D, from Buffalo after t hours. In this equation, the 59 represents the
(2) speed of the car. Every hour, the car is 59 miles closer because it is traveling at 59 miles per hour.
10. Faith wants to use the formula C(f) = (5/9) (f - 32) to convert degrees Fahrenheit, f, to degrees Celsius, C(f). If Faith calculated C(68), what would her result be?
(1) 20^{o} Celsius. First of all, you are converting from Fahrenheit to Celsius, so choices (2) and (4) are right out. Second, when it comes to regular temperatures (not extreme cold), Celsius are lower than Fahrenheit. (They are the same at 40 below zero!) So you didn't even have to do the math to answer this. If you've been to the Caribbean in the winter, it's been in the 20s there and it's beach weather!
If you didn't realize that, there's the mathematical answer:
11.Which scenario represents exponential growth?
(3) A species of fly doubles its population every month during the summer. Doubling is exponential. The rest are linear.
12.What is the minimum value of the function y = |x+ 3| - 2?
(1) -2. The vertex of this absolute value graph is (-3, -2). The minimum value is -2.
13.What type of relationship exists between the number of pages printed on a printer and the amount of ink used by that printer?
(2) positive correlation, and causal. The more pages printed, the more ink is used. And one causes the other.
And non-causal positive relationship would be something like "ice cream sales go up and swimsuit sales go up". Both go up in the warmer weather.
14.A computer application generates a sequence of musical notes using the function f(n) = 6(16)^{n}, where n is the number of the note in the sequence and f(n) is the note frequency in hertz. Which function will generate the same note sequence as f(n)?
(2) h(n) = 6(2)^{4n}. Forget everything in this question about music. You don't need to know it.
What you need to know is that 6(16)^{n} = 6(2)^{4n}.
16 is the same as 2^{4}
So (16)^{n} is the same as (2^{4})^{n}
Rules of exponents say that you multiply the 4 and the n, which gives (2)^{4n}.
15. Which value of x is a solution to the equation 13 - 36x^{2} = - 12?
(4) -5/6
You can rewrite the equation as 36x^{2} - 25 = 0, which is a Difference of Squares.
That factors into conjugates: (6x + 5)(6x - 5). For either binomial to equal zero under the
Zero Product Property, x would have to equal positive or negative 5/6.
16.Which point is a solution to the system below?
(4) (-3,2)
Look for a point that fits both inequalities. You can plug the points into both equations, or you can graph the lines in your graphing calculator.
1/2 is NOT less than -6(1) + 4
6 is NOT less that -6(0) = 4
5 IS less than -6(-1/2) + 4, which is 7, BUT 2(5) = 10 is NOT less than -12(-1/2) + 4, which is also 10. (They're equal.)
Choice (4) is left. 2 IS less than (-6)(-3) + 4 which is 22 AND 2(2) = 4 IS less than (-12)(-3) + 4, which is 40.
17.When the function f(x) = x^{2} is multiplied by the value a, where a > 1, the graph of the new function, g(x) = ax^{2}
(2) opens upward and is narrower. Because a > 1, it is positive, so the parabola opens upward. When a > 1, the parabola shoots up faster, and as a result, it will be narrower.
18.Andy has $310 in his account. Each week, w, he withdraws $30 for his expenses. Which expression could be used if he wanted to find out how much money he had left after 8 weeks?
(4) 280 - 30(w - 1). This is a silly answer, but it is mathematically correct.
The answer you would expect is 310 - 30w, because 30 is subtracted each week.
First notice that only two choices have w multiplied by 30, and one of them is adding money.
After 1 week, 310 becomes 280. The number of weeks after that is reduced by 1, so w - 1.
19.The daily cost of production in a factory is calculated using c(x) = 200 + 16x, where x is the number of complete products manufactured. Which set of numbers best defines the domain of c(x)?
(4) whole numbers. Complete products would have to be positive integers or zero, which is the set of whole numbers. Negative numbers and fractions (real and rational numbers) wouldn't make sense.
20. Noah conducted a survey on sports participation. He created the following two dot plots to represent the number of students participating, by age, in soccer and basketball .
Which statement about the given data sets is correct?(4) The data for basketball players have a greater mean than the data
for soccer players.
Without doing the calculations, you can see that the data for the soccer players skew left and while the data for the basketball players skew right. So the mean and the median are both going to be higher for the basketball players. (You can check the median by counting the dots quickly enough if you don't believe me.)
21.A graph of average resting heart rates is shown below. The average resting heart rate for adults is 72 beats per minute, but doctors consider resting rates from 60-100 beats per minute within normal range.
Which statement about average resting heart rates is not supported by the graph?(1) A 10-year-old has the same average resting heart rate as a 20-year-old.
According to the graph, a 10-year-old's rate is higher (approximate 90 beats/min) than a 20-year-old (72 beats/min).
22.The method of completing the square was used to solve the equation 2x^{2} - 12x + 6 = 0. Which equation is a correct step when using this method?
(1) (x - 3)^{2} = 6
To complete the square, first divide all the terms by 2 to get rid of the leading coefficient. Next, look at the middle term, which in this case is now 6. Half of 6 is 3, and 3 squared is 9. We need to have 9 as the constant to have a perfect square. To get 9, we have to add 6 to both sides of the equation. (I sometimes teach students to subtract 3 and then add 9 if they can't figure out the square that they need. It's an extra step, but they can get the correct solution, so it's worth it.)
Once we have x^{2} - 6x + 9, we can factor that into (x - 3)(x - 3) or simply (x - 3)^{2}, which is the square we wanted to complete. Hint: Once we knew that half of 6 is 3, we also knew that (x - 3)^{2} would be in our answer.
23.Nancy works for a company that offers two types of savings plans. Plan A is represented on the graph below.
Plan B is represented by the function f(x) = 0.01 + 0.05x^{2}, where x is the number of weeks. Nancy wants to have the highest savings possible after a year. Nancy picks Plan B.correct, because Plan B is a quadratic function and will increase at a faster rate .
First, x^{2} is quadratic, not exponential. Second, check to make sure that choice (3), "Plan A will have a higher value after 1 year" is incorrect. Although the function will increase at a faster rate, it starts off with a much lower rate. Check that it will catch up.
f(52) = 0.01 + 0.05(52)^{2} = 135.21. On the graph, Plan A gives an amount between $100 and $110 at 52 weeks.
24. The 2014 winner of the Boston Marathon runs as many as 120 miles per week. During the last few weeks of his training for an event, his mileage can be modeled by M(w) = 120(.90)^{w-l}, where w represents the number of weeks since training began. Which statement is true about the model M(w)?
(3) M (w) represents the total mileage run in a given week.
The number of miles is decreasing by 10% per week, or is 90% of the previous week. This may sound counterproductive in training if you don't know that a marathon is only about 26 miles. It is stated in the question that w is the number of weeks since training began and not the number of weeks left until the marathon.
End of Part I
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When an incorrigible punster meets an unmovable girlfriend, something's gotta give.
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This question was worth 6 credits
37.Ian is borrowing $1000 from his parents to buy a notebook computer. He plans to pay them back
at the rate of $60 per month. Ken is borrowing $600 from his parents to purchase a snowboard.
He plans to pay his parents back at the rate of $20 per month.
Write an equation that can be used to determine after how many months the boys will owe the same
amount.
Determine algebraically and state in how many months the two boys will owe the same amount.
State the amount they will owe at this time.
Ian claims that he will have his loan paid off 6 months after he and Ken owe the same amount.
Determine and state if Ian is correct. Explain your reasoning.
First part, the two equations:
Ian: y = 1000 - 60x, where x is the number of months and y is the remaining balance owed.
Ken: y = 600 - 20x.
They will be equal at 1000 - 60x = 600 - 20x
Second part: Solve the equation. Note, if you're equation is incorrect, solve it anyway because you will still get some credit!
Third part: Solve Ian's equation for y = 0, when no money is owed.
Alternatively, six months after 10 is 16 months,
and y = 1000 - 60(16) = 1000 - 960 = 40. He will still owe $40 at 16 months, so he is not paid off.
End of Part IV
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Typos happen.
A true story. Happened last week. Could NOT have planned it that way.
No idea when they're fixing the copier, but in the meantime, I need to edit and update that worksheet so the graph is ALWAYS in the same position.
And, no, I didn't want to use anyone else's meme generators nor anyone else's pictures.
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33. Graph f(x) = | x | and g(x) = -x^{2} + 6 on the grid below.
Does f(-2) = g(-2)? Use your graph to explain why or why not.
Note that if you explain using substitution instead of graphing, you will not get credit.
The graph looks like this:
So f(-2) = g(-2) because f(-2) = 2 and g(-2) = 2. The two graphs intersect at the point (-2, 2).
34. Two friends went to a restaurant and ordered one plain pizza and two sodas. Their bill totaled
$15.95. Later that day, five friends went to the same restaurant. They ordered three plain pizzas
and each person had one soda. Their bill totaled $45.90.
Write and solve a system of equations to determine the price of one plain pizza.
[Only an algebraic solution can receive full credit.]
Let p = the price of one pizza and let s = the price of one soda. The system of equations would then be:
You can solve this system using elimination by multiplying the first equation by -3, or by substitution if you solve the first equation for p by subtracting 2s from both sides.
Elimination first:
Substitution:
Yes, that is an unusual amount for a pizza, but they didn't want to make it "obvious". $1.95 is a reasonable price for a soda.
35. Tanya is making homemade greeting cards. The data table below represents the amount she spends in dollars,f(x), in terms of the number of cards she makes, x.
You need to find the slope and the y-intercept, which means the rate of change and the initial value.
Pick any two points on the table. Let's say you hate decimals, so you pick (6, 9) and (10, 12). Rate of change is the difference of the y values divided by the difference of the x-values. That is, (12 - 9) / (10 - 6) = 3 / 4 or .75. (You might want to use the decimal for the calculator in the next step.)
Next, you need the y-intercept. Start with y = mx + b and plug in values for x, y and m:
The function is f(x) = .75x + 4.5
You need to explain the answer in context of the problem. $4.50 is the initial value, the starting cost for making the cards, and $0.75 is the cost per card.
36. Alex launched a ball into the air. The height of the ball can be represented by the equation h = -8t^{2} + 40t + 5, where his the height, in units, and t is the time, in seconds, after the ball was launched. Graph the equation from t = 0 to t= 5 seconds.
The graph will look like this:
To find the height at t=2.5, substitute into the equation, and enter it into your calculator:
h = -8(2.5)^{2} + 40(2.5) + 5 = 55
The vertex is (2.5, 55). At 2.5 seconds the ball reaches its highest point, which is 55 feet in the air.
End of Part III
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Comments, questions, corrections and concerns are all welcome.
Typos happen.
Tell me about it! The first 1100+ comics are the hardest!
The wurst thing about this: at first, I thought, I can't do this. But then the song came into my head and the words wrote seemed to write themselves ... until I tried writing them down and I realized I didn't have as much as I thought. Do you know why?
Because my muse is a brat!
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