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Posts filed under: Math

Now that the Daily PWN email list has been going for a while and I’ve got some good data on the questions, I thought I’d compile a list of the ones people are missing most frequently. If you’re looking for a quick skill sharpening on some tough problems, why not give these a try?

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Here’s another Proving Grounds installment! The aim of the following five-question quiz is to work your graphing calculator muscles, so my recommendation is that you try to solve them by graphing even if your first inclination would be to solve them another way. My solutions for this drill will be entirely calculator-based; spend enough time…

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It’s been a while since we did one of these! The following five-question quiz (all about a histogram, by special request) will be available to everyone for one week, and then it will only be available to registered Math Guide Owners. (If you don’t have a Math Guide, now is a pretty good time to…

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This is not really SAT specific or even particularly SAT useful, but I made the video above to help you create a basic quadratic formula program for your TI graphing calculator if that’s a thing you’d like to do.

If you’ve programmed things into your calculator before and don’t feel like watching a whole video, you can also just enter the program below. Make sure you’re careful with your quotation marks and parentheses, and always test the program with multiple quadratics to make sure it’s always giving you correct answers before you use it for anything important.

:Disp "AX2+BX+C=0"
:Prompt A
:Prompt B
:Prompt C
:Disp "ROOTS:"
:Disp (-B+√(B2-4AC))/(2A)
:Disp (-B—√(B2-4AC))/(2A)

 

The new SAT requires you to know a number of special equation forms—to know which one you need to use in a given situation, and to know how to get into that form if it’s not the one you’re given by using algebraic manipulation. Some equation forms (vertex form of a parabola and the standard circle equation immediately spring to mind) contain binomial squares, e.g. (x+1)^2, as essential ingredients. To get a non-standard equation into these forms, you’ll often have to complete the square. I know, I know, you’ve done this a million times in school. Still, I often find students haven’t done this in a long time and need a little bit of a refresher. So here we are.

First, the equations in question.

Vertex form of a parabola: y=a(x-h)^2+k, where the vertex of the parabola is at (h,k).

Standard circle equation: (x-h)^2+(y-k)^2=r^2, where a circle with radius r has its center at (h,k).

Say you’re given a parabola that’s not in vertex form and you need to put it in vertex form. How do you do that?

No calculator; grid-in

y=x^2-8x+6

The parabola formed when the equation above is graphed in the xy-plane has its vertex at (a,b). What is the value of a-b ?

Completing the square isn’t the only way to solve this question, but I’d argue it’s the fastest. All we need to do to go from the given form to the vertex form is figure out which binomial square the x^2-8x part of the equation is the beginning of. With practice, this becomes second nature and you probably won’t need the rule, but the rule is that x^2+b is the beginning of \left(x+\dfrac{b}{2}\right)^2.* In this case, that means that x^2-8x is the beginning of (x-4)^2.

Now, what do you get when you FOIL out (x-4)^2? You get x^2-8x+16. That’s not what we have above—we have x^2-8x+6 instead. Luckily, we can do anything we want to the right side of the equation provided that we keep the equation balanced by doing the same thing to the left, so we can just add 10 to both sides!

y=x^2-8x+6

y+10=x^2-8x+6+10

y+10=x^2-8x+16

From there, we’re almost done. Now we can convert the right side to the binomial square we wanted, and then get y by itself again to land in vertex form.

y+10=(x-4)^2

y=(x-4)^2-10

So, there you have it: the parabola in question has a vertex of (4,-10). Since the question said the vertex was at (a,b), we know that a=4, b=-10, and a-b=4-(-10)=14. So, 14 is the answer.

Let’s practice with a few more, shall we? Try to do the following drill without a calculator. All three questions are grid-ins.

1.
y=x^2-12x+33

The parabola formed when the equation above is graphed in the xy-plane has its vertex at (a,b). What is the value of a+b ?

Question 1 of 3

2. When the equation y^2=(x+3)(-x+5) is graphed in the xy-plane, it forms a circle. What is x-coordinate of the center of the circle?

Question 2 of 3

3. What is the radius of the circle with equation x^2+y^2+6x-10y=2 ?

Question 3 of 3


 

 
 
 
 
 
 
* I’m intentionally limiting this post to scenarios where the leading coefficient in the square being completed is 1. So far, I have not seen an official question of this type where that is not the case.

I’m back after a hiatus with another Proving Grounds Quiz. Usual Proving Grounds rules apply: this quiz is open to everyone for a week, but then it’s only open to Math Guide owners. Good luck! *Data source: City of Bridgeport Office of Policy and Management. Accessed 2015-06-14 at http://www.bridgeportct.gov/content/89019/96401/default.aspx…

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I’m back with another Proving Grounds quiz. These quizzes are available to everyone for one week, and then they’re only available to Math Guide owners. Want to join the swelling ranks of the PWN Army of Math Guide owners? You can buy the guide directly from me through the PWN store, or grab it on…

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As I did for the last iteration of the SAT, I’ve been collecting the explanations I write on my Q&A sites for Official Test questions in a Google Spreadsheet for easy reference. The new test is still new, so I haven’t been asked MOST of the questions yet, but I figure it’s time to get this page out into the world. If you’re working through the official SAT practice tests and you have a sneaking suspicion that the official explanation is unnecessarily complicated, well, then here’s a way to get a second opinion.

PS: Download the Official Tests here.

Last week was a no-calculator installment of the Proving Grounds—this week it’s all grid-ins! Remember, if you want to access previous Proving Grounds quizzes, or if you want to be able to access this and future ones after they’ve been up for a week, all you have to do is be a Math Guide owner. You…

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Another Proving Grounds quiz coming your way. This one you should do without your calculator. Remember, Proving Grounds quizzes are available to everyone for one week, and then only available to Math Guide owners. Not a Math Guide owner yet? Got $20? :)…

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Time for another Proving Grounds installment, folks. Remember, these quizzes are available for one week for everybody, and then they’re for Math Guide owners only. “How can I get to be a Math Guide owner?” you ask?! Well, either you buy one right from me, or you forward me your receipt from some place like…

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Here’s the second installment of the Proving Grounds quizzes, which are available for everyone to try for one week before they become exclusive to Math Guide owners. A little parabola-heavy this week, but then again, who doesn’t love parabolas? Good luck! Mechanical Calculators image By Ezrdr (Own work) [CC BY-SA 3.0], via Wikimedia Commons…

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Below is the first in a series of tough quizzes for the new SAT that will be available to everyone for a week, and then only available to Math Guide owners. If you want access to all these quizzes long term, well, you might want to grab a Math Guide. They’re on sale now, and…

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If you live on the East Coast, then chances are pretty good that your January 23rd SAT just recently became your February 20th SAT. That’s a bummer, no? Then again, maybe you’re happy—you’ve just been given a whole extra month to study. My suggestion: take advantage.

I was planning on taking my Math Guide and Essay Guide out of print today, but now that they’re relevant for a small group of you for another month, I’ve just slashed their prices instead. If you don’t have my books, yet, and you’ve just found out you have more time than you thought to get ready, I invite you to grab them at your favorite online retailer at a deep discount. Here they are at Amazon—my favorite online retailer.

   

I’ve received a few questions in my email asking about the Beta I’m running (full info here). Specifically, people want to know what they get if they sign up.

I figured that the best way to answer that question is just to post one of the chapters up here in public. If you join the Beta, you get access to chapters of the new book as I finish them just like you’ll see below. You also get the opportunity to, if you spot a typo, submit it to me in exchange for a $5 gift card.

Registered Math Guide owners can already access the Beta for free. For everyone else, it’s $16.99. When the actual book is released, paying Beta customers will be able to buy it at a big discount.