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 II
25. Graph the function y = -SQRT(x + 3) on the set of axes below.
Put the equation into a graphing calculator, or make a table of values. The negative sign tells you that the graph will be inverted (upside down). The +3 under the radical sign tells you that the graph will shift three spaces to the left. Any value of x that results in a negative number under the radical can be ignored -- those values are NOT in the domain of this function. The smallest possible values is -3 because -3 + 3 = 0, and the square root of 0 is 0.
Your graph should look like the one below. Be sure to graph the following points: (-3, 0), (-2, -1), (1, -2) and (6, -3).
26. Richard is asked to transform the graph of b(x) below.
The -2 inside the parentheses will shift the graph two places to the right. The -3 outside the parentheses will shift the graph three places down.
So the point (0, -2) will shift to (2, -5).
27. Consider the pattern of squares below
Which type of model, linear or exponential, should be used to determine how many squares are in the nth pattern? Explain your answer.
The first pattern has 2 squares, the next has 4, the one after has 8. The rate of change is not constant, so it is not linear. However, there is a constant ratio, each pattern is twice as big as the one before it, so the pattern is exponential.
You didn't need to give the function but it would be something like f(n) = 2n.
28. When multiplying polynomials for a math assignment, Pat found the product to be -4x + 8x2 - 2x3 + 5. He then had to state the leading coefficient of this polynomial. Pat wrote down -4. Do you agree with Pat's answer? Explain your reasoning.
First, a reminder: You will NOT score a point for answering just "Yes" or "No" to any question like this. HOWEVER, you WILL LOSE a point if you don't include "Yes" or "No" in the answer. In this case, the answer is "No".
The leading coefficient is the coefficient of the term with the highest exponent because in standard form the term with the highest exponent comes first, and then the term with the second highest, etc.
In standard form, this expression is -2x3 + 8x2 - 4x + 5. Note that the sign goes with the term the follows it. Also, for clarity when reading, the negative in front of the 4x became a minus sign. It could have been written as + (-4x). Likewise, the minus in front of 2x3 became a negative sign for the same reason.
The leading coefficient for this expression is -2.
For full credit, answer "No, I don't agree" and explain that in standard form the term with the highest coefficient is the leading term, so the leading coefficient is "-2" because -2x3 is the term with the highest coefficient.
29. Is the sum of 3*SQRT(2) and 4*SQRT(2) rational or irrational? Explain your answer.
Sorry for the lack of radical signs on my keyboard.
Each of the two expressions is irrational because they cannot be written as fractions. The sum of two positive irrational numbers is always irrational.
I added positive because there is one exception to the rule that the sum of two irrational numbers is always irrational: if the numbers are opposites, then the sum is zero, which is rational. But it is obvious that they are not opposites.
Another way to state it is that the sum of the two numbers is 7*SQRT(2), which is irrational because the product of a rational number and an irrational number is irrational.
Sue wrote her answer in point-slope form and Kathy wrote her answer in slope-intercept form. However, you still have to prove that they both have the correct answer. You can calculate that the slope is (4 - 1) / (-3 - 6) = 3 / (-9) = -1/3, which is in both equations.
Sue used the slope -3 and the point (-3, 4) to write her equation.
If you distribute the (-1/3) on the right side, and then add 4 to both sides to isolate the y, you will get Kathy's equation.
If this didn't occur to you, but you checked both points in both equations and found out that they all worked, you should get full credit.
30. The graph below shows two functions, f(x) and g(x). State all the values of x for which f(x) = g(x).
-3 and 1.
Do not give the points of intersection. They are only asking for the x values.
I'll be honest with you: I don't like this question. It feels like they should be asking for more, but they aren't. They only want the two x-values, and each one is worth 1 point.
If you give the coordinates you lose a point. So if your entire answer is (1, 3), you scored zero.
31. Find the zeros of f(x) = (x - 3)2 - 49, algebraically.
You can put this in your graphing calculator to find the answers in advance, if you like. And those answers will be worth 1 of the 2 points.
Algebraically, set the function equal to 0:
(x - 3)2 = 49
take the square root of both sides -- don't forget to use +/-
x - 3 = +/- 7
split the equation into two parts to finish, one with +, one with -
x - 3 = 7 or x - 3 = -7
x = 10 or x = -4
32. Solve the equation below for x in terms of a.
Solve for x in terms of a means that you have to isolate x on the left side of the equation and have the numbers, operations and the letter a on the right side. You will not get a numeric answer.
4ax + 12 - 3ax = 25 + 3a
ax + 12 = 25 + 3a
ax = 13 + 3a
x = (13 + 3a)/a
Do NOT cancel out the "a" characters. You can't; it doesn't work like that.
How did you do? Any questions?
I hope you did well because, overall, they weren't difficult question (in my opinion). They could have been a little trickier, even for just two points.
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