What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.
August 2016 Algebra Regents, Part I
1. The graph below shows the distance in miles, m, hiked from a camp in h hours.
Which hourly interval had the greatest rate of change?
(1) hour 0 to hour 1. The line is the steepest between 0 and 1 hour, which means is has the greatest slope, or the greatest rate of change. In the first hour, 2 miles were hiked, then only 1.5 miles, then 1 mile, then 0.5 miles. In the last hour, the hikers did not move away from camp at all.
2. The solution of an equation with two variables, x and y, is
(3) the set of all ordered pairs, (x, y), that make the equation true. That's the definition of a solution to an equation. It is not about solving for the zeroes, as in choices (1) and (2).
3. Which statistic can not be determined from a box plot representing the scores on a math test in Mrs. DeRidder's algebra class?
(4) the score that occurs most frequently. You cannot tell the mode from a box plot (also known as a box and whisker plot). You can only see the quartiles, including the minimum, median and maximum.
4. Which chart could represent f(x) = -2x + 6?
(4). You don't need to calculate anything. You can figure this one out logically. When x = 0, f(x) must equal 6. Only choices (1) and (4) have (0, 6) for (x, f(x)). The rate of change is -2, but choice (1) has a positive rate of change, so it is incorrect. Choice (4) is the only one left.
If you put y = -2x + 6 into your graphing calculator and check the table, you will see that (4) has the correct ordered pairs.
5. If f(n) = (n - 1)2 + 3n, which statement is true?
(2) f(-2) = 3. You can skip (1) because in this function a positive number can't give a negative result. (-2 - 1)2 + 3(-2) = (-3)2 - 6 = 9 - 6 = 3.
6. The table below shows 6 students' overall averages and their averages in their math class.
If a linear model is applied to these data, which statement best describes the correlation coefficient?
(2) It is close to 1. You don't need to put this in your calculator. (You could if you truly weren't sure. When the top row is higher, the bottom is higher. When it is lower, the bottom is lower. There is a positive correlation, so choices (1) and (3) are out. Furthermore, the pairs are very close together, suggesting a strong correlation, which means that it would be closer to 1 than it would be to 0.5.
7. What is the solution to 2h + 8 > 3h - 6?
(1) h < 14.
Subtract 2h from both sides, you get 8 > h - 6
Add 6 to each side, you get 14 > h
Turn it around, h < 14
8. Which expression is equivalent to 36x2 - 100?
(2) 4(3x + 5)(3x - 5). You can tell by looking at it that it will be a Difference of Squares problem because it has no middle term. That means that the two factors will be conjugates; that is, the binomials will be the same except that one will have addition and one will have subtraction. That difference is what causes the middle terms to cancel out. So (1) and (3) are immediately eliminated.
3x times 3x is 9x2, which when multiplied by 4 has a product of 36x2, as given in the question. However, 9x times 9x is 81x2, which is already too big. Choice (4) is eliminated.
9. Patricia is trying to compare the average rainfall of New York to that of Arizona. A comparison between these two states for the months of July through September would best be measured in
(3) inches per month. This is a strange question. It assumes you know something about rainfall in Arizona. (Because it is a New York state exam, it's fair to assume that you know something about the rainfall in New York -- you live here.)
There isn't a lot of rainfall in Arizona, even when there may be in New York. Measuring the rainfall by the hour is just silly -- it doesn't rain for that many hours in an entire summer, so the ratio would be a tiny, tiny decimal. Rain, like snow, is usually measured in inches, not feet. It is not uncommon to hear forecasts of 10-14 inches of snow, as opposed to 1-2 feet of snow. Even when talking about the entire accumulation for the season, you are more likely to hear, say, 24 inches, rather than 2 feet.
So for a bunch of reasons, a few actually having to do with math, the best answer is inches per month.
10. Which function defines the sequence -6, -10, -14, -18, ..., where f(6) = -26?
(1) f(x) = -4x - 2. The sequence is going down by four. Only one choice, (1), has -4x in it.
Check: -4(6) - 2 = -24 - 2 = -26. Yes.
11. Which function has the greatest y-intercept?
(4) (the graph). The y-intercept of the graph is 5. In choice (1), 3(0) = 0, which is less than 5. In choice (2), if 2(0) + 3y = 12, then y = 4, which is less than 5. In choice (3), if there is a point on a line at (1, -4) and the slope is 2, then subtract 2 from the y value and you find that the line crosses the y-axis at (0, -6), which is less than 5.
12. What is the product of 2x + 3 and 4x2 - 5x + 6?
(3) 8x3 + 2x2 - 3x + 18. Notice that the first and last terms are the same for all four choices, so we don't have to worry about those.
(2x)(-5x) = -10x2 and (3)(4x2) = 12x2, -10 + 12 = -2. so we can eliminate choices (1) and (2).
(2x)(6) = 12x and (3)(-5x) = -15x, 12 - 15 = -3. So the choice is (3).
13. The height of a rocket, at selected times, is shown in the table below.
Based on these data, which statement is not a valid conclusion?
Based on these data, which statement is not a valid conclusion?
(3) The rocket was in the air approximately 6 seconds before hitting the ground. It was still above the ground after 7 seconds. The other choices give information that can be found in the table. It was at 180 feet at 0 seconds (launch); it hit is maximum at 3 seconds; it was above the 300 feet for 2 seconds from about t = 2 to t = 4.
14. A parking garage charges a base rate of $3.50 for up to 2 hours, and an hourly rate for each additional hour. The sign below gives the prices for up to 5 hours of parking.
Which linear equation can be used to find x, the additional hourly parking rate?
(3) 2x + 3.50 = 14.50. Four hours cost $14.50, which is the first 2 hours plus 2 additional hours. The first 2 hours = $3.50 and the additional 2 hours are 2x.
Note that this is just one possible equation. A simpler equation would've been x + $3.50 = 9.00, but that was not an option.
15. Which function has a constant rate of change equal to -3?
(4) 2y = -6x + 10. Put the equation into slope-intercept form by dividing both sides by 2, and you get y = -3x + 5, which has a slope (rate of change) of -3.
In choice (1), the rate of change is increasing. In choice (2), the slope is -2. Choice three is not a linear function, so it does not have a constant rate of change, even though the change from x = 1 to x = 2 is, in fact, -3, but this does not continue.
16. Kendal bought x boxes of cookies to bring to a party. Each box contains 12 cookies. She decides to keep two boxes for herself. She brings 60 cookies to the party. Which equation can be used to find the number of boxes, x, Kendal bought?
(3) 12x - 24 = 60. If x is the number of boxes, then 12x is the number of cookies altogether. If she kept 2 boxes, then she took away 24 cookies leaving 60.
17. The table below shows the temperature, T(m), of a cup of hot chocolate that is allowed to chill over several minutes, m.
Which expression best fits the data for T(m)?
(1) 150(0.85)m. The hot chocolate is getting cooler, and the temperature is getting lower. Therefore, the base must be less than one (exponential decay), so choices (2) and (4) are out. If m = 0, then 150(0.85)0 = 150(1) = 150, which fits the table. The correct choice is (1). In choice (3), m - 1 would give an exponent of -1, which would actually increase the temperature at m = 0.
18. As x increases beyond 25, which function will have the largest value?
(1) f(x) = 1.5x. Exponential functions with bases greater than 1 will climb higher at a faster rate than quadratic or cubic functions. You can put all four of these functions in your calculator and see the results for 25 and greater.
19. What are the solutions to the equation 3x2 + 10x = 8?
(1) 2/3 and -4. You can enter y = 3x2 + 10x - 8 into your calculator and search for the zeroes of the function. Or you can check the numbers 2, -2, 4, and -4, in whatever order you like, to see which one makes the equation true. Because no number repeats, you don't need to worry about substituting the fractions into the equation.
20. An online company lets you download songs for $0.99 each after you have paid a $5 membership fee. Which domain would be most appropriate to calculate the cost to download songs?
(2) whole numbers greater than or equal to one. Rational numbers include fractions, and you can't download half a song (realistically, and financially, speaking). Integers less than zero are negative, which is also impossible. Whole numbers less than or equal to one means 1 and 0, only, which means you could only download at most 1 song ever.
21. The function f(x) = 3x2 + 12x + 11 can be written in vertex form as
(3) f(x) = 3(x + 2)2 - 1. Once again, you can put all the functions in your calculator and see which two are the same. You can save a little work if you realize that you need to divide 12 by 3 before you halve it -- so the parentheses have to say "(x + 2)" not "(x + 6)", and you can eliminate choices (1) and (2).
There is a quicker way for this particular problem if you see the relationship in the numbers -- but that is something you develop from doing a bunch of these kinds of problems, so I'll do the longer way first.
f(x) = 3x2 + 12x + 11, subtract 11 from both sides
f(x) - 11 = 3x2 + 12x, factor the 3 on the right side
f(x) - 11 = 3(x2 + 4x), half of 4 is 2, (2)2 = 4, (3)(4) = 12
.... so add 12 to both sides of the function
f(x) + 1 = 3(x2 + 4x) + 12, divide the 12 by 3 to get +4 inside the parentheses to complete the square
f(x) + 1 = 3(x2 + 4x + 4), factor the polynomial into a binomial squared
f(x) + 1 = 3(x + 2)2, subtract 1 from both sides to isolate f(x)
f(x) = 3(x + 2)2 - 1.
My "shorter" method comes from recognizes that 3 goes into 12 and if you add 1 more to 11 you get 12. If you factor 3 from 3x2 + 12x + 12, you get 3(x2 + 4x + 4), which you should recognize as the perfect square of (x + 2). And because 1 was added, 1 has to be subtracted from the final expression.
22. A system of equations is given below.
2x + y = 4
(4) x + 2y = 5, 4x + 2y = 12. If you double the second equation, you should have 4x + 2y = 8, not 12. In the other three choices, one of the equations is multiplied correctly.
23. based on the graph below, which expression is a possible factorization of p(x)?
(1) (x + 3)(x - 2)(x - 4). There are three zeroes, so there are three factors, eliminating choices (3) and (4). The zeroes are -3, 2, and 4, so the factors are the opposites (x - (-3)), (x - 2), (x - 4).
24. Milton has his money invested in a stock portfolio. The value, v(x) of his portfolio can be modeled with the function v(x) = 30,000((0.78)x, where x is the number of years since he made his investment. Which statement describes the rate of change of the value of his portfolio?
(2) It decreases 22% per year. The base is less than 1, so it is decay, decreasing. The rate it is decreasing is 1.00 - .78 = .22. Each year, he loses 22% of his money, and still has 78% of it remaining.
To be edited so I can add all of the images as time permits.
How did you do? Any questions?