Daily Regents: Literal Equations (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 34

34. The formula for the area of a trapezoid is A = (1/2)h(b₁ + b₂).Express b₁ in terms of A, h, and b₂.

The area of a trapezoid is 60 square feet, its height is 6 ft, and one base is 12 ft. Find the number of feet in the other base.

You want to isolate the variable we want to solve for, in this case b₁:

A = (1/2)h(b₁ + b₂)
Multiply by 2: 2A = h(b₁ + b₂)
Divide by h: 2A/h = b₁ + b₂
Subtract b₂ from both sides: (2A/h) - b₂ = b₁
b₁ = (2A/h) - b₂

To find the answer to the second part, substitute the values you have into the equation you just created, and evaluate.

b₁ = (2A/h) - b₂ = (2(60)/6) - 12 = 120/6 - 12 = 20 - 12 = 8
The other base is 8 ft.

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.

Ken-Do #11: Memorial Day

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

All gave some. Some gave all.

I did try longer, more complicated set-builder notations, but scrapped them.

Oddly, this is comic #1111 or 11-11, which would be Veterans Day, which is totally coincidental. I didn't go out of my way to make this happen. If anything, I've been wanting to update the page for at least a week now. Real world stuff.

Happy Memorial Day




Come back often for more funny math and geeky comics.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Solution

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

y = mx + earl




Come back often for more funny math and geeky comics.

Daily Regents: Rectangular Garden

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 34

34. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters, as shown in the diagram below. Together the walkway and the garden have an area of 396 square meters.

Write an equation that can be used to find x, the width of the walkway.

Describe how your equation models the situation.

Determine and state the width of the walkway, in meters.

Unlike the previous question, this was a majorly complicated problem, with more binomial multiplication.

The length of the rectangle is x + 16 + x or (2x + 16).
The width of the rectangle is x + 12 + x or (2x + 12).
The area of the rectangle is 396.

So (2x + 16)(2x + 12) = 396
4x² + 24x + 32x + 192 = 396
4x² + 56x - 204 = 0
Any time you have a coefficient in front of x², check if you can factor it out. In this case, all the coefficients are multiples of 4, so we can just divide each term on the left by 4. Dividing 0 by 4 will leave 0 on the right side.
x² + 14x - 51 = 0 Factor the trinomial back into two binomials and apply the Zero Product Property. (One of the binomials MUST equal zero.)
(x + 17)(x - 3) = 0
x + 17 = 0 or x - 3 = 0
x = -17 or x = 3

We can discard the negative value not because we made a mistake, but because a negative value makes no sense in the context of length and width.
You are left with the width of the walkway being 3 meters wide.

The equation models the situation because the Area is the product of the length times the width. The length of the garden plus the walkway on either side is 2x + 16. The width of the garden plus the walkway is 2x + 12. The Area of the garden is 396 square meters.

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.

Daily Regents: Trinomials

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 33

33. Write an equation that defines m(x) as a trinomial where m(x) = (3x - 1)(3 - x) + 4x² + 19. Solve for x when m(x) = 0.

You need to multiply the binomials and then combine all the like terms.
(3x - 1)(3 - x) = 9x - 3x² - 3 + x = -3x² + 10x - 3
To that, add 4x² + 19.
You get m(x) = x² + 10x + 16. Don't forget to write it as an equation!

You needed to find when m(x) = 0, so x² + 10x + 16 = 0. That's a simple quadratic equation to solve. What two factors of 16 add up to 10? That's 8 and 2.
So (x + 8)(x + 2) = 0
x + 8 = 0 or x + 2 = 0
Then x = -8 or x = -2.

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.

Daily Regents: Box-and-Whisker Plot

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 32

32. Robin collected data on the number of hours she watched television on Sunday through Thursday nights for a period of 3 weeks. The data are shown in the table below.
Using an appropriate scale on the number line below, construct a box plot for the 15 values.

You have two choices.

First, you can put the data into order and then find the Five-Number Summary: the minimum, Q1, median, Q3, and maximum.

The numbers in order are: 1. 1.5, 1.5, 2, 2, 2.5, 2.5, 3, 3, 3, 3.5, 4, 4, 4.5, 5.
The median is the middle number. There are 15 numbers and the 8th is the middle. That is 3
Q1 is the middle of the lower seven numbers. The 4th number is 2.
Q3 is the middle of the upper seven numbers. The 11th number is 4.
The minimum is the 1st (lowest) number, 1. The maximum is the last (highest) number, 5.

So the Five-Number Summary, and the numbers that will be used to make the Box-and-Whisker Plot, are 1, 2, 3, 4, and 5. (Note: It will NOT always work out so nicely.)

Make a scale on the number line. There is enough room to number it from 0 to 5, counting by 1/2. (That is, every two spaces is 1.) You could use a scale of 1, if you like, but your graph will be smaller than mine. That's okay.

Put a point above each number in our Five-Number Summary: 1, 2, 3, 4, 5.
Draw a box around the middle three with a line through the median.
Draw whiskers connecting the first two points and the last two points.
You're done.

The second method is to use the Graphing Calculator. Take a look at the 10 figures in the diagram below. Don't let the 10 screens frighten you.

If you use this method, make sure you copy some of the information onto the test so you will get points for showing work. In particular, the Five-Number Summary would be good.

To use the calculator, press the STAT key. You should see Figure 1. You need to EDIT a List -- fill it with the data that you will be using. Press (1). Figure 2 shows some of the data that was entered into the first list, L₁. Enter all 15 numbers. Now you are ready to plot you graph.

Press STAT again. When you get the menu, hit the right arrow to get to the CALC menu. See 1-Var Stats? That's what we want: 1 variable statistics. Press Enter twice.

You will get a scrollable screen of information. Scroll down to see the Five-Number Summary (Figure 6).

Press 2nd and Y= to get to the STAT PLOT screen (Figure 7) and press ENTER to bring you to the Plot page (Figure 8). Make sure that ON is highlighted, and arrow over to the icon that looks like a box-and-whisker plot and press ENTER.

If you press Graph, your graph won't appear because the window is formatted from -10 to 10 on the x-axis. You need to adjust this. Press ZOOM and then press 9: ZoomStat. Your box-and-whisker plot will appear. Put the graph on the paper, using the points that the 1-Var Stats function gave you.

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.

Daily Regents: Functions and Factoring

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 30 and 31

30. The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}. Could f be represented by {(1, 2), (3, 4), (5, 6), (7, 2)}? Justify your answer.

I'm still not happy about the wording of this one. I've been told that I'm "splitting hairs", but, in my opinion, the question is asked backward.

The question is asking if the set given could be function f. In other words, is it a function and does it have the correct domain and range?

Let's answer that question.

The set of ordered pairs is a function because no x value is mapped to more than one y value. The domain (the x values) are {1, 3, 5, 7} as in f. The range (the y values) are {2, 4, 6} which matches the range of f. So, yes, it could represent f.

Remember that "yes" or "no" without an explanation is worth ZERO points.

31. Factor the expression x⁴ + 6x² - 7 completely.

Usually, a “factor completely” question has a Greatest Common Factor (GCF) component to it. This one doesn't, but it does have an extra step that isn't obvious at first.

If you are thrown off by the exponents being 4 and 2, instead of 2 and 1, that's okay. They will act pretty much the same way.

x⁴ + 6x² - 7 factors into (x² + 7) (x² - 1). You now have two binomials, each with an x² term. If you look closely, you can see that we aren't finished. We can factor further.

Using the Difference of Squares Rule (x² - 1) can be factored into (x – 1)(x + 1).

This makes the final answer:

(x² + 7) (x – 1)(x + 1)
Note that (x² + 7) has no real roots and cannot be factored further.

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.

Daily Regents: Graphing a Linear Equation

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, Question 29

29. On the set of axes below, draw the graph of the equation y = -(3/4)x + 3.
Is the point (3,2) a solution to the equation? Explain your answer based on the graph drawn.

Unlike the previous question, the graph is definitely required. You can use your graphing calculator to get the Table of Values, or you just plot the points directly using the slope and the y-intercept.

The y-intercept is 3, so the first point is (0, 3). The slope is -3/4, which means that for every four places to the right you go, the next point will be three places down. Also, for every four places to the left, the next point will be three places up.

The next point will be at (4, 0), then (8, -3). Working the other way, there are points at (-4, 6) and (-8, 9). Draw a straight line through these points. If they don't line up, you miscounted somewhere.

Don't forget to answer and explain the second part. No explanation means loss of a point.

You need three things to score two points on this question. The graph, "No" and an explanation. If you make a mistake with the graph, you will lose one point and the rest of the answer must match your graph. (That is, if by some chance, your incorrect graph goes through the point (3, 2), then you must answer "Yes" to the second question.)

Point (3, 2) is not a solution because it is not a point on the line. The points on the line are all the solutions.

If you want to use the graphing calculator, look at the images below. In Figure 1, the parentheses around the fraction are not necessary, but the are a good habit to get into if you have this version of the operating system. (Newer models will show the fractions.) You get to this screen using the Y= button.

Figure 2 shows how the graph will look. And Figure 3 shows you some of the Table of Values. (Press "2nd" and "Graph" to get this screen.) Plot the points that have whole number values.

Notice that (3, 2) is not in the Table of Values. However, you have to give your answer in reference to the line, not the table.

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.

Daily Regents: Translating a Function (June 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.

June 2014, Question 28

The graph is not required, but it could have been used instead to explain the answer.

The vertex of the parabola represented by f(x) = x² - 4x + 3 has coordinates (2, -1). Find the coordinates of the vertex of the parabola defined by g(x) = f(x - 2). Explain how you arrived at your answer.
[The use of the set of axes below is optional.]

When translating a function, the minus two inside the parenthesis will shift all the points two places to the right. So the vertex (2, -1) in f(x) will be translated to (4, -1) in g(x).

Graphically, it looks like this. You can use the graph with the explanation or alone. However, if you only use the graph, you MUST label the vertex because that is the point that was asked for, along with an explanation. (You made a graph, that shows how you arrived at your answer. A table of values would help also.)

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.

Sunny Day

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

One of these days, those glasses are going to give me away!




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Daily Regents: Exponential Functions and Inequalities

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 26 and 27

These two half-page questions are short and easy to explain, so we'll do both of them today.

26. The breakdown of a sample of a chemical compound is represented by the function p(t) = 300(0.5)^t, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

This is a definition question. There is no solving involved, but you do you have answer referencing the question.

The 300 represents the original amount of the chemical compound.
The 0.5 represents the rate of decay of the compound.

27. Given 2x + ax - 7 > -12, determine the largest integer value of a when x = -1.

Substitute -1 for x:

2(-1) + (-1)a - 7 > -12 Combine like terms: -a - 9 > -12 Add 9 to both sides: -a > -3 Multiply by -1 and don't forget to flip the inequality symbol: a < 3

The largest integer value less than 3 is 2.

Any questions?

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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.

It's Not the Planning

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(C)Copyright 2016, C. Burke.

The best-made lesson plans are only good until the class begins.




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Daily Regents: Graphing a Square Root Function

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, Question 25

Draw the graph of y = SQRT(x) - 1 on the set of axes below.
As show in the picture, SQRT means "square root", but I can't type that. However, keep that function in mind when we get to the calculator.

Here is what you saw on the paper:

On your graphing calculator, press the "Y=" key. Look at Figure 1 below.
Press "2nd" and "x²" to get the square root symbol and the open parenthesis.
Press the variable and then don't forget to close the parentheses! If you don't close them, then the "- 1" will be under the radical, and that is NOT what you want.
Hit GRAPH and you'll get Figure 2. This is what you need to graph on your paper.

The easiest way to get the values you need is to check the Table of Values. "2nd" and "Graph". Figure 3 and Figure 4 show you are of the coordinate pairs from 0 to 10. Notice that -1 and -2 are ERROR because they are out of the domain of the function. You won't have any line on the left side of the graph.

Plot the four points that have whole number values and draw a curve through them. Do not connect the four point with three straight lines.

Your final answer should look like this:

Any questions?

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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.

Happy Mother's Day 2016

(come back soon)

Algebra/Geometry Tutoring and Regents Prep in Brooklyn

Final exams will be here before you know it, and soon after will be the Regents exams.

I have been teaching in NYC high schools and grading these exams for 15 years. If anyone in Brooklyn needs a tutor for the Regents, whether Common Core or the older version, you can contact me via email, mrburkemath@gmail.com.

I'm available most weekday afternoons at this point. Reasonable rates.

Grams

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(C)Copyright 2016, C. Burke.

I can't say I've made a habit of drawing histograms with logarithmic scales. Can't remember when else I might have done even one.




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