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.
Part II is posted here.
June 2016 Algebra Regents, Part I
1. The expression x4 - 16 is equivalent to
(3) (x2 + 4)(x2 - 4). Difference of Squares Rule
Additionally, a +4 times -4 gives you - 16, and there's only one choice.
If you were really stuck, you could have multiplied all the choices or put them in your graphing calculator and looked for a match
2. An expression of the fifth degree is written with a leading coefficient of seven and a constant of six. Which expression is correctly written for these conditions?
(4) 7x5 + 2x2 + 6. "Fifth degree is written with a leading coefficient of seven" means 7x5.
Also, only one choice had a constant of 6.
3. The table below shows the year and the number of households in a building that had high-speed broadband internet access. (table omitted)
For which interval of time was the average rate of change the smallest?
(1) 2002-2004. Average rate of change is (difference is number of households) / (difference in years).
In each case, the number of years is 2, so you only need to find the smallest numerator.
23 - 11 = 12. 33 - 16 = 17. 42 - 23 = 19. 47 - 33 = 14.
Choice 1 is smallest.
4. The scatterplot beow compares the number of bags of popcorn and the number of sodas sold at each performance of the circus over one week. (image omitted)
Which conclusion can be drawn from the scatterplot?
(2) There is a positive correlation between popcorn sales and soda sales. As one rises, so does the other.
5. The Celluloid Cinema sold 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold $1470 worth of tickets, which system of equation could be used to determine how many adult tickets, a, and how many child tickets, c, were sold?
(1) a + c = 150; 10.25a + 7.75c = 1470. The number of adult (a) + the number of child (c) tickets is (=) 150.
The amount of money from adult tickets is 10.25 times a. The amount of money from child tickets is 7.75 times c.
6. The tables below (image omitted) show the values of four different functions for given values of x.
Which table represent a linear function?
(1) f(x). A linear function will have a constant rate of change, slope, common difference, etc. 19-12 = 7, 26-19 = 7, 33-26 = 7 (x increases by 1 in each case).
7. The acidity in a swimming pool is considered normal if the average of three pH readings, p, is defined such that 7.0 < p < 7.8. If the first two reading are 7.2 and 7.6, which value for the third reading will result in an overall rating of normal?
(2) 7.3. Here is why Number Sense is important. If all three numbers are between 7.0 and 7.8, then the average must be as well. Common sense says that 7.3 will result in an average between those two extremes.
That being said, an average is found by adding the three numbers and dividing by three (the number of data items).
The averages are 7.0, 7.4, 7.8, 7.9. Only the second one is within the compound inequality.
Calculator trick: 7.2 + 7.6 = 14.8, right? Try this in your graphing calculator:
The parentheses and the brackets are important. No spaces after the commas.
8. Dan took 12.5 seconds to run the 100-meter dash. He calculated the time to be approximately
(1) 0.2083 minute. The 100 is irrelevant. They didn't ask about his speed, just his time. To convert seconds to minutes, divide by 60.
12.5 seconds is less than one minute so 750 minutes is silly (and really slow: 6 1/2 hours to finish the race!)
Also, .2 and .5 hours are well over one minute, so they can be crossed out.
9. When 3x + 2 < 5(x - 4) is solved for x, the solution is
(4) x > 11.
3x + 2 < 5x - 20
2 < 2x - 20
22 < 2x 11 < x
x > 11
If you graphed y = 3x + 2 and y = 5(x - 4), you would have found the intersection point at x = 11, which, in this case, would have eliminated 3 choices.
10. The expression 3(x2 - 1) - (x2 - 7x + 10) is equivalent to
(2) 2x2 + 7x - 13. 3x2 - x2 = 2x2, which is in all the choices.
0 - (-7x) = +7x, which eliminates (1) and (3).
3(-1) - 10 = -13. Choice (2).
11. The range of the function f(x) = x2 + 2x - 8 is all real numbers
(2) greater than or equal to -9. Range is the y values on the graph. You can put this in your graphing calculator and see the answer.
Or you can find the vertex. X = -b / (2a) = -2/(2*1) = -1, and f(-1) = (-1)2 + 2(-1) - 8 = -9.
The parabola opens up, so the vertex is a minimun and all other f(x) must be higher than that.
12. The zeroes of the function f(x) = x2 - 5x - 6 are
(1) -1 and 6. Again, put it in your calculator and look at the graph and the Table of Values. Or factor it.
(x - 6)(x + 1) = 0
x = 6 or x = -1
13. In a sequence, the first term is 4 and the common difference is 3. The fifth term of this sequence is
(3) 16. Either write f(n) = 3(n - 1) + 4, f(5) = 3(5-1) + 4 = 16, or just write the sequence: 4, 7, 10, 13, 16.
14. The growth of a certain organism can be modeled by C(t) = 10(1.029)24x, where C(t) is the total number of cells after t hours. Which function is approximately equivalent to C(t)?
C(t) = 10(1.986)t.
First of all, the graph is exponential growth. Two of the choices have a base of .083, which is exponential decay. So choices (1) and (2) are out.
At this point, you can put the original equation and choices (3) and (4) into your graphing calculator.
Looking at choice (3), the only differences are that the base has changed and the exponent is t instead of 24t. That means that if 1.029 to the 24th power is 1.986, then this is the correct answer. If you put that in your calculator, you see that it is the same.
15. A public opinion poll was taken to explore the relationship between age and support for a candidate in an election. The results of the poll are summarized in the table below. (Table omitted).
What percent of the 21-40 age group was for the candidate?
(4) 60. The entire age group is 30 + 12 + 8 = 50 people. "For" was 30 out of 50, and 30/50 = .6 = 60%.
16. Which equation and ordered pair represent the correct vertex from and vertex for f(x) = x2 - 12x + 7?
(3) f(x) = (x - 6)2 - 29. First, (x - 6) means that h, the x-coordinate of the vertex, is 6, so eliminate choices (2) and (4).
Next, either complete the square or put the equations in the graphing calculator! Seriously.
To complete the square do the following:
f(x) + 36 = x2 - 12x + 36 + 7
f(x) + 36 = (x - 6)2 + 7
f(x) = (x - 6)2 - 29
17. A student invests $500 for 3 years in a savings account that earns 4% interest per year. No further deposits or withdrawals are made during this time. Which statement does not yield the correct balance in the account at the end of 3 years?
(2) 500(1 - .04)3. It should be obvious that choice (2) shows decay and not growth so it cannot be the answer.
Choice (1) is the standard way to write it. Choice (3) is what you get if you expand the exponent of 3 -- three factors of (1 + .04). Choice (4) is what you get if you use the simple interest formula and add the interest year after year.
18. The line represent by the equation 4y + 2x = 33.6 shares a solution point with the line represent by the table below. (Table omitted)
The solution for this system is
(4) (6.0, 5.4). First off, eliminate choices (2) and (3). The y values 5.0 and 4.6 are in the table with different x values, so neither can be the intersection point.
Next, you need to find the equation of the line. At this point, AGAIN, I will remind you that you can use your calculator to give you the equation. Put two sets of points into Lists and do a Linear Regression to get the equation of the line. y = .2x + 4.2.
Check .2(-14) + 4.2 = 1.4, no good, and .2(6) + 4.2 = 5.4, which checks.
If you didn't do a linear regression, first find the slope of the line:
Using (2, 4.6) and (4, 5): (5 - 4.6) / (4 - 2) = .4 / 2 = .2, the slope of the line is .2, so y = .2x + b.
Next find b: 5 = .2(4) + b
5 = .8 + b
b = 4.2
So y = .2x + 4.2
Now check which choices work. If one choice works, double check that it works for 4y + 2x = 33.6.
19. What is the solution of the equation 2(x + 2)2 - 4 = 28?
(3) 2 and -6. Test-taking Strategy: Try 6 first: Put 2(6 + 2)2 - 4 in the caculator. It does not equal 28. Cross out (1) and (4). You know now that "2" has to be a solution, so you only need to check (-6).
2(-6 + 2)2 - 4 = 28. Check.
Or use the graphing calculator! Subtract 28 from both sides and you have 2(x + 2)2 - 4 - 28 = 0.
Graph y = 2(x + 2)2 - 32 and look for the zeroes!
20. The dot plot below represents the number of pets owned by students in the class
Which statement about the data is true?
(3) The mean is 3. If you analyze the data, there are 20 dots. The 10th and 11th piece of information are both 3, so the median is 3. Split the data and find Q1 and Q3 (which are, basically, the median of the lower half and the median of the upper half). Q1 = 2 and Q3 = 4, so the interquartile range is 4 - 2 = 2. Choices (1) and (2) are eliminated.
An outlier is a piece of data so far away from the rest of the data that it is discounted (maybe it was a mistake in data entry?) so that it doesn't sway (or "skew") the results. There are no points way off (say, at 10 or 11). So choice (4) is eliminated. That leaves choice (3).
If you add up the values represented by each of the 20 dots and then divide by 20, you will get 2.75, not 3.
Once again, you can use the calculator. If you use list L1, you can enter the 20 pieces of data: 0, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, and do 1-Var Stats. You win get the mean, Q1, median and Q3. (You can then calculate IQR.) And you can see the box-and-whisker plot, if you wanted to, and Trace the points.
21. What is the largest integer, x, for which the value of f(x) = 5x4 + 30x2 + 9 will be greater than the value of g(x) = 3x?
(3) 9. Calculate the values of f(x) and g(x) for 7, 8, 9 and 10.
Or put the equations in the graphing calculator and see which is the last time f(x) is bigger by checking the Table of values. (The graph is really big. You won't see it by looking at the graph itself.
22. The graph of the functions f(x) = |x - 3| + 1 and g(x) = 2x + 1 are shown. Which statement about these functions is true?
(2) The solution to f(x) = g(x) is 1. When x = 1, f(x) = 3 and g(x) = 3. The solution is the value of x that makes it happen, not the function values.
You may have noticed that choices (1) and (4) essentially say the same thing.
23. A store sells self-serve yogurt sundaes. The function C(w) represents the cost, in dollars, of a sundae weighing w ounces. An appropriate domain for the function would be
(4) nonnegative rational numbers. Negative ounces do not make sense. (Zero ounces cost zero, so that's okay.) You don't have to buy an integer value of yogurt in your sundae. It would be difficult to imagine how they could scoop it out exactly.
24. Sara was asked to solve this word problem: "The product of two consecutive integers is 156. What are the integers?"
What type of equation should she create to solve this problem?
(2) quadratic. "The product of two consecutive numbers" would be represented by (N)(N + 1), which will create a quadratic equation.
End of Part I
Who did you do? Any questions?