Monday, May 30, 2016

Daily Regents: Graphing and Translating Absolute Value (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 33

33. On the axes below, graph f(x) = |3x|.
If g(x) = f(x) - 2, how is the graph of f(x) translated to form the graph of g(x)?
If h(x) = f(x - 4), how is the graph of f(x) translated to form the graph of h(x)?

You can use the graphing calculator to see what the function looks like. Press the MATH key, and then arrow over to the NUM menu and abs( will be the first option. Depending on your calculator's operating system, you will either get abs( written as a function, or you will see the symbols: | |

If you have the symbols, put the variable between them. If you have the function, add the variable and then close the parentheses! This will be important later in the problem.

g(x) will translate the function 2 spaces down. If you subtract two from a number after taking the square root, the value will always decrease by 2. Example: f(0) = 0, g(0) = 0 - 2 = -2.

h(x) will translate the function 4 spaces to the right. Taking the absolute value after subtracting means that any negative results will become positive. Example f(0) = 0, h(0) = f(0-4) = 4, but f(4) = 4, h(4) = f(4-4) = 0. The vertex of the function has shifted four places to the right.

You can look at the graphs of g(x) and h(x) in the calculator to compare.
Remember: if your calculator's operating system shows abs( for absolute value, you must close the parentheses:
y = abs(x) - 2
y = abs(x - 4)




Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Sunday, May 29, 2016

Daily Regents Review: Pinning the Posts

Due to family obligations, it looks like there won't be a Sunday update.

However, in order to keep the "Daily" tag going, I pinned the earlier posts on Pinterest:

https://www.pinterest.com/cjburke23/algebra-regents-review/

Bookmark it, and you will be able to find all the questions later on.

Saturday, May 28, 2016

Daily Regents: Completing the Square (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 32

32. A student was given the equation x2 + 6x - 13 = 0 to solve by completing the square. The first step that was written is shown below.

x2 + 6x = 13

The next step in the student’s process was x2 + 6x + c = 13 + c
State the value of c that creates a perfect square trinomial.
Explain how the value of c is determined.

To complete the square, you have to take one-half of the "b" term, which is 6. That gives you 3. Then you have to square the 3, which gives you 9.
Therefore, c = 9.

(x + 3)2 = x2 + 6x + 9


Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Friday, May 27, 2016

Daily Regents: Scatter Plot Residuals (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 31

31. The table below represents the residuals for a line of best fit.
Plot these residuals on the set of axes below.
Using the plot, assess the fit of the line for these residuals and justify your answer.

Plot the points on the table. It should look like this:

If a regression line is a "good fit", the residuals should be scattered about, above and below the x-axis, randomly.
If a pattern emerges, then the regression is not a good fit for the data. (For example, suppose you did a linear regression, but the data suggested that you should have done a quadratic regression instead. Then a pattern will emerge in the residuals.)

The graph below shows that there is a pattern among the residuals, and it looks like it forms a curve.
DO NOT draw the curve! Just explain that it is not a good fit because of the pattern.




Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Thursday, May 26, 2016

Daily Regents: Inequalities (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 30

30. Solve the inequality below to determine and state the smallest possible value for x in the solution set.

3(x + 3) ≤ 5x - 3

Use the Distributive Property to get rid of the parentheses, then solve it like an equation with variables on both sides:

3x + 9 ≤ 5x - 3
9 ≤ 2x - 3
12 ≤ 2x
6 ≤ x

Therefore, x > 6
The smallest possible value is 6.




Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Wednesday, May 25, 2016

Daily Regents: Maximum Values (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 29

29. Let f be the function represented by the graph below. (See the image above). Let g be a function such that g(x) = -1/2 x2 + 4x + 3.
Determine which function has the larger maximum value. Justify your answer.

The vertex of f(x) is (1, 6), so the maximum value of f is 6.

To find the vertex of g(x), first let's find the Axis of Symmetry, which runs through the Vertex. The formula for that is
x = -b / 2a
x = -(4) / ((2)(-1/2)). Note:: if you put this in your calculator, you need that "extra" pair of parentheses around the denominator.
x = -4 / -1 = 4

Find the value of g(4). This will be its maximum height. g(4) = -1/2(4)2 + 4(4) + 3 = 11

Therefore, g(x) has a higher maximum value than f(x).


Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Tuesday, May 24, 2016

Daily Regents: Functions and Multiplying Polynomials

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 27 and 28

27. Guy and Jim work at a furniture store. Guy is paid $185 per week plus 3% of his total sales in dollars, x, which can be represented by g(x) = 185 + 0.03x. Jim is paid $275 per week plus 2.5% of his total sales in dollars, x, which can be represented by f(x) = 275 + 0.025x. Determine the value of x, in dollars, that will make their weekly pay the same.

To find the value of x where both functions are the same, you can set them each to each other, and solve the equation.

g(x) = f(x)
185 + 0.03x = 275 + 0.025x
185 + 0.005x = 275
0.005x = 90
x = 90/0.005 = 18,000

If they both have sales of $18,000, their salaries will be equal.

If you want to check, put 18,000 into each function. Both will give you $725.

28. Express the product of 2x2 + 7x - 10 and x + 5 in standard form.

Use the Distributive Property. Then combine like terms, writing the terms starting with the highest exponent and going in descending order.

(x)(2x2 + 7x - 10) + (5)(2x2 + 7x - 10)
2x3 + 7x2 - 10x + 10x2 + 35x - 50
2x3 + 17x2 + 25x - 50



Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Monday, May 23, 2016

Daily Regents: Factoring and Compound Interest

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

August 2014, Questions 25 and 26

25. In the equation x2 + 10x + 24 = (x + a)(x + b), b is an integer. Find algebraically all possible values of b.

x2 + 10x + 24 factors into (x + 4)(x + 6), the order of the two binomials doesn't matter.
So the possible values of b are 4 and 6.

26. Rhonda deposited $3000 in an account in the Merrick National Bank, earning 4.2% interest, compounded annually. She made no deposits or withdrawals. Write an equation that can be used to find B, her account balance after t years.

The formula you need is B = P(1 + r)t
P = 3000, the principal, or starting amount
r = .042, which is 4.2% as a decimal
t is time. No specific value is given, so use the variable.

That makes the answer B = 3000(1.042)t


Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Sunday, May 22, 2016

Daily Regents: Quadratic-Linear Systems of Equations

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 37

37. A company is considering building a manufacturing plant. They determine the weekly production cost at site A to be A(x) = 3x2 while the production cost at site B is B(x) = 8x + 3, where x represents the number of products, in hundreds, and A(x) and B(x) are the production costs, in hundreds of dollars.

Graph the production cost functions on the set of axes below and label them site A and site B.

State the positive value(s) of x for which the production costs at the two sites are equal. Explain how you determined your answer.

If the company plans on manufacturing 200 products per week, which site should they use? Justify your answer.

It is confusing that everything is listed in hundreds, but it's okay, because they basically cancel out as far as the scale is concerned. Also a headache is that each box on the vertical side represents 2, so all the odd numbers are going to be between the lines.

You can put both of these functions into your graphing calculator and get the points you need.

If you don't know how to find them on the calculator, you can make two Tables of Values, as follows:

Plot each set of points. You can see my results below. (On my graph, I also plotted (.5, 1.5) so I could draw the curve better in my graph. You wouldn't be expected to.)

The two questions from the following page assume that you did the graph. However, they can be solved algebraically, without the graph. This is a bit more work, but it can be done. When they say to Explain your answer, you can refer to the graph, if you drew the graph. If you didn't, you have to show the computations you did to get the answer. Otherwise, it's assumed that you're just taking a guess. Any guess that's 50-50 (choice A or choice B) is NOT worth any credit.

The positive value of x is 3, which represents 300 units. (Note: x is NOT 300.)
They ask for the positive value, because if you solve algebraically, you may get a negative value as well, which would be discarded.

If only 200 units were made, the graph shows that site A is cheaper than site B.
Without a graph, you would have to calculate A(2) = 3(2)2 = 12 and B(2) = 8(2) + 3 = 19 to determine that A is cheaper.




Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Saturday, May 21, 2016

Daily Regents: Modeling Systems of Equations

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 36

36. An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday.

Write an equation to represent to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat's numbers possible? Use your equation to justify your answer. Later, Pat found a record sowing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?

This is a lot of work for a system of equations.

We'll use C for the number of cats and D for the number of dogs. That makes 2.35C the amount spent on cats and 5.50D is the amount spent on dogs.

The equation for Wednesday is

2.35C + 5.50D = 89.50.

For the second part, you want to see if Pat's numbers could be correct. The way to do that would be to put them into the equation and see if it works. Substitute 8 for C and 14 for D, and check if it's a true statement.

2.35(8) + 5.50(14) = 89.50 ?
18.80 + 77.00 = 89.50 ?
95.80 =/= 89.50 X

This is NOT a possible combination.

The final part states that there were 22 cats and dogs altogether. So the total number of cats (C) plus the number of dogs D is 22. You can write that as an equation

C + D = 22.

We now have a system of equations that we can solve simultaneously.

Multiply this second equation by -2.35 to match the coefficient of C in the first equation. You can solve using Elimination.

-2.35C + -2.35D = -51.70
2.35C + 5.50D = 89.50
combine the two equations: 3.15D = 37.80
divide by 3.15: D = 12

There are 12 Dogs, so there are 22 - 12 = 10 = C, 10 Cats.


Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Friday, May 20, 2016

Daily Regents: Functions (Movie Rentals)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 35

35. Caitlin has a movie rental card worth $175. After she rents the first movie, the card's value is$172.25. After she rents the second movie, its value is $169.50. After she rents the third movie, the car is worth $166.75. Assuming the pattern continues, write an equation to define A(n), the amount of money on the rental card after n rentals.

Caitlin rents a movie every Friday night. How many weeks in a row can she afford to rent a movie using the rental card only? Explain how you arrived at your answer.

First, "Explain" means in words; equations will NOT be enough information. I am not kidding.

Subtract 175-172.25 and you get $2.75. Check the next and the next to be sure, but you see that each rental is $2.75
That makes the equation A(n) = -2.75n + 175.

To answer the second part, make an inequality greater than 0, and find the largest whole number which makes it true. (Or make an equation equal to 0 and drop everything after the decimal.)


-2.75n + 175 > 0
-2.75n > -175
n < 63.63

Each movie costs $2.75 to rent. If you divide $175 by $2.75, you can get 63 rentals. There is money left over, but not enough for another rental using the card only.

At one rental per week, she can do this for 63 weeks before she runs out of money.


Any questions?


If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


Thursday, May 19, 2016

Solution

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

y = mx + earl




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