Monday, May 11, 2015

January 2015 Common Core Algebra Regents, Part II

Update: I now have a Common Core Regents Review books available on Amazon.

If you missed the previous post, here is Part I.

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The Common Core exams have more open-ended questions, so I will be splitting them up into two posts. Here are (most of) the Part II questions.

25. Ms. Fox asked her class “Is the sum of 4.2 and the (square root of 2) rational or irrational?” Patrick answered that the sum would be irrational. State whether Patrick is correct or incorrect. Justify your reasoning.

4.2 is a rational number and the square root of 2 is an irrational number. The sum of a rational number and an irrational number is always irrational.

The square root of 2 is 1.414213562… which continues forever without repeating. If you add 4.2, you get 5.614213562… which still continues forever without repeating.

26. The school newspaper surveyed the student body for an article about club membership. The table below shows the number of students in each grade level who belong to one or more clubs.

If there are 180 students in ninth grade, what percentage of the ninth grade students belong to more than one club?

There are a total of (33+12=) 45 ninth grade students who are in more than one club. Divide 45/180 = .25 and multiply by 100%. The answer is 25%.

27. A function is shown in the table below.

If included in the table, which ordered pair, (-4,1) or (1, -4), would result in a relation that is no longer a function? Explain your answer.

A relation cannot be a function if there are two separate output values for the same input. The relation contains (-4, 2). If you include (-4, 1) then there will be two outputs for the same input.

This is not the case for (1, -4). There is no x value of 1 in the table yet.

28. Subtract 5x2 + 2x – 11 from 3x2 + 8x – 7. Express the result as a trinomial.

The word from tells you that the second polynomial is the top one in the subtraction problem.

The answer is -2x2 + 6x + 4.

29. Solve the equation 4x2 – 12x = 7 algebraically for x.

If you put y = 4x2 – 12x – 7 into your graphing calculator and find the zeroes, you will not get full credit.

Subtract 7 from both sides of the equation and you get

4x2 – 12x – 7 = 0

You could use guess and check to factor the binomial, knowing that the factors of 7 are only 1 and 7, and the factors of 4 are either 1 and 4 or 2 and 2. There aren’t too many options, so it isn’t a bad possibility – as long as the factors are rational.

The other method is to use the quadratic formula.

The two methods are shown below:

Using either method the solution set is {-0.5, 3.5}

30. Graph the following function on the set of axes below.

This is a piecewise function. Literally, it is a piece of one function and a piece of a different function. The inequalities listed are the domains for each portion of it. In other words, which x-values do you need to use for that part of the graph.

Read the piecewise function like this:

F of x equals the absolute value of x WHEN x is greater than or equal to -3 and less than 1. It is equal to 4 WHEN x is greater than or equal to 1 and less than or equal to 8.

You will only use values between -3 and 8. Do not graph any other values.
The output will look like this:

Note that you need to include the open circle and the closed circle because a function cannot have two y values for any x value. The open circle indicates a boundary that is not part of the solution.

31. A gardener is planting two types of trees: Type A is three feet tall and grows at a rate of 15 inches per year. Type B is four feet tall and grows at a rate of 10 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.

First, you have to fix your units. The problem uses both feet and inches. You have to convert to one of them. Converting to inches is easier and won’t require decimals.

Three feet is 36 inches. Four feet is 48 inches.
So Type A grows at: y = 36 + 15x
And Type B grows at: y = 48 + 10x

They will meet at 36 + 15x = 48 + 10x. Subtract 10 from both sides.
That gives you 36 + 5x = 48. Subtract 36 from both sides.
That gives you 5x = 12. Divide by 5.
And x = 12/5 years, which is 2.4 years, which is reasonable because plants grow continuously. They don’t just shoot up 10 or 15 inches at the end of the year.

That’s it for now. Questions 32-37 will be in the next installment.

6 comments:

Anonymous said...

For guestion 28 you are wrong

(x, why?) said...

Thank you. The typo has been corrected. Should've been a plus sign, not a minus.

(But, then, you knew that, didn't you?)

Anonymous said...

how do you label the lines on the piecewise graph?

(x, why?) said...

You don't need to. It's a single graph, spread out across multiple domains.

I'm looking online, and I don't see examples where they are labeled. The rubric for the question does not mention labeling it.

Anonymous said...

Late but what's x and y for question #31 ? �� please and thank you.

(x, why?) said...

It says it in the description that x = 2.4 years.

You don't need to find y for the problem, but if you substitute 2.4 for x in either (or both) of the functions, you will get 72.