Parabolas

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I recently came across a problem that required knowledge of parabola formulas. That surprised me. Although parabola problems show up relatively frequently, they usually require little more than logic. However, I thought that provided a nice opportunity to refresh on all things parabola.

A parabola is defined mathematically by this formula: y= ax^2 + bx + c. We see parabolas in nature most often when we look at projectiles, like a cannonball shot out of a cannon or a jump shot out of the hand of Steph Curry. Parabolas are generally u-shaped and are symmetrical about the vertex, which is either the highest or lowest point of the parabola, depending on the orientation.

Whether our parabola is cupped upward or downward is determined by the sign of the “a” term in the formula we saw above. When a is positive, the parabola will have a vertex at the bottom and open upward. When a is negative, the parabola will have a vertex at the top and open downward.

However, there is more than one way to define a parabola mathematically. We can also solve a parabola if we have the vertex and another point on the parabola. We do that by using the similar formula y = a(x-h)^2 + k. The coordinates of the vertex are (h,k).

So using that information, find the equation of a parabola with vertex (-2,1) containing the point (1, 19). The first thing we need to do is solve for a by inserting our points into the formula. We get:

19 = a(1-(-2)^2 + 1

19= a(3^2) + 1

19= 9a + 1

Now, we put our a into the formula with our vertex (h,k), but instead of using the x and y from a specific

point we’re going to solve for the generic x and y.

y= 2(x-(-2)^2 + 1

y= 2(x+2)^2 +1

Now expand:

y=2(x+2)(x+2) +1

y=2x^2 + 8x + 9

Now we’ve solved for the equation of this parabola and we could mathematically figure out all of the

points on this curve.

I hope that’s been a good refresher on parabolas!

 Graduation Attitude

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It’s getting to be the graduation time of year when my Facebook feed is filled with smiling happy pictures of my former students who have gone on to earn college or graduate degrees. It’s wonderful to know that I was able to have a part in these stories, but more than that it offers perspective.

Ask any of those graduates what the biggest obstacle on the way to her degree was, and she won’t tell you it was the SAT or the GRE or the GMAT. That was just a first step. A stepping stone obstacle on the way to bigger and tougher challenges. Those who will be successful with those later obstacles, exhibit five key attitudes that help them achieve.

1. Be Positive- Sure skepticism is “cool” but a positive attitude breeds success. People that believe they can achieve something special are most often the ones who do.

2. Set Goals- It’s a long way between here and graduation. If you don’t set markers along the way to gauge your progress, it’s tough to succeed.

3. Build a Support System- When the going gets tough– whether now or later– you need people who care about you to be there. Whether that’s someone who will lend a sympathetic ear, help you work through a tough problem, or just take you out for ice cream and a break, successful people don’t do it alone.

4. Take the Long View- Minor disappointments are going to pop up along the way. Will you dwell on them, or will you shift your focus toward making the best of what’s ahead?

5. Keep Going- No matter what keep going. Don’t stop, and you will achieve your goals.

Now get to it and get back to studying!

 Reading Comprehension: Theory and Practice

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In my latest post I suggested an approach to reading comprehension that asks you to simplify concepts to basic level so that you can catch the core of the arguments that are most essential for answering questions. That’s fine in theory, but theory and practice do not always equate so I wanted to take this opportunity to look at how that strategy might work in practice.

I have always suggested reading The Economist to my graduate-level students looking for material that mimics reading passages on the GRE or GMAT. High school students may find those articles a level above what’s likely to appear on the ACT or SAT, but there is still some value there. So, I went to the front-page of the magazine’s website looking for the article that would seem most intimidating to students who jumped into it without any introduction on a test. Here is the article that I found. Open it in another tab and read it and then come back.

Welcome back. Unless you’re a big beetle aficionado you probably weren’t familiar with the subject of that article. No worries. Let’s look at how we can break it down into easily digestible pieces that will get to the core of the passage.

Paragraph 1: A mystery is presented: why so many beetles?

Paragraph 2: Previous attempts to figure this out focused on new species. A new approach looks at whether beetle species are less likely to go extinct.

Paragraph 3: How they tested theory: fossils

Paragraph 4: More methods and conclusion: specific group seems hard to exterminate

Paragraph 5: Other beetles have died off but this specific group seems to survive

Paragraph 6: This isn’t a complete answer to the question, but it’s a start

The basics I’ve presented above distill the article into it’s essential structure and basic elements. My outline may not answer every question you’re asked, but it will answer many of them and it will point you toward the answer in most other cases. There are lots of places in this article to get tripped up. Did you know what “catholic tastes in food” were? I didn’t. I was able to figure it out from context, but unless it was asked about specifically, it’s certainly not something you should focus on. The same goes for the specific research methodology or the names or the particular scientists involved in the research. Rather than over-complicating passages, simplify them down like you would explain them to a 12-year old. By focusing on the core ideas, you’ll get to the heart of what you’re likely to be questioned on.

 Reading Comprehension for Too Smart People

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If you’ve been in a bookstore in the past twenty years, you’ve certainly seen the ubiquitous “For Dummies” series. You can learn about car repairs, tennis or the newest programming language as long as you’re willing to put up with a small insult to your intelligence. The brilliance of the brand has always been that the amount of knowledge you’re assumed to have coming in is very low, so if you’re new to the concept you can pick up the basics before moving onto progressively more difficult topics.

There is a beauty in hearing an elegantly simplified concept, and the skill of turning complex ideas into simple thoughts is in high demand. It’s in especially high demand on the reading comprehension section of your test. You’re presented with a passage that you probably couldn’t care less about, and asked to answer a series of questions about it. There’s probably a whole lot of background and underlying facts behind the passage you read, but you aren’t responsible for that information. You only need to know what’s on the page.

Unfortunately, our tendency when reading unfamiliar passages is to flaunt our depth and mastery of the material when speaking to others about it. No one wants to admit that he doesn’t understand much of the text he’s reading, so he makes assumptions to fill the gaps in his knowledge, or masks his explanations in the jargon he’s reading but doesn’t quite understand.

To succeed in reading comprehension you must resist those urges. Instead, you must seek to simplify. Maybe not a dummy, but imagine you’re explaining what you’re reading to someone who is 12 or 13 years old. They are not as wise or sophisticated as you are so you’ll have to break down concepts carefully. You’ll have to slowly walk through pieces generally rather than being over-specific. You’ll do your best to make complicated ideas less complicated. This process, rather than blunting your understanding of the text will actually help you focus much better on the core ideas of the passage which are crucial to reading comprehension success.

In reading comprehension, it’s always important not to be too smart for your own good.