The post GRE Coordinate Geometry Shortcut: No Graphs! appeared first on Magoosh GRE Blog.

]]>For many, coordinate geometry is already a daunting concept. When a question dispenses with the graph all together, students can feel even more at a loss. If you fall into this group, do not despair. Here is a helpful guideline:

This advice may seem counterintuitive. After all, the problem didn’t provide a graph. Wouldn’t the first step be to graph the problem out?

Many coordinate geometry concepts sans graph are testing your conceptual thinking. Take the follow problem:

1. Which of the following lines do not contain coordinate points that are both negative?

**Solution:**

The slope formula is important – if the question is explicitly asking for the slope. What is often more important is knowing that a line with a negative slope – from left to right – slopes downward. A positive slope, unsurprisingly, slopes upward.

Think of it this way – start at a negative x-coordinate (say -2) of a line. If you were to place a ball on the line would roll down the line as it moved into positive territory for the x-coordinate? If so the line is negative, if not the line is positive.

For this problem, we are looking for a line that does not pass through the third quadrant – the quadrant in which x and y are both negative. Graph the point (-2, -2). That’s in the third quadrant.

Now here’s the big conceptual part – any line that slopes upwards will always pass through Quadrant III. Graph it if you have to – or simply imagine a line of infinite length sloping upwards. Anyway you try to do so there will always be the Third Quadrant waiting to claim a part of your line.

Now, imagine a downward sloping line. Is it also crossing through the third quadrant? Well, move the entire line to the right. At a certain point, your line will no longer be in the Third Quadrant. As long as that line crosses the y-axis at a positive value, it will never cross through the Third Quadrant.

Now you only need to find two things: a line that has a positive y-intercept and a negative slope. And that is much better than having to graph every one of the equations in answer choices A – E!

Only answer (C) x + y = 2, which can be re-written as y = -x + 2, has a negative slope (-1) and positive y-intercept (+2).

If a coordinate geometry question does not provide a graph, it is often testing conceptual thinking. So unless you are really desperate (which can happen on the GRE), and have time to spare, avoid graphing and think conceptually.

If you are unsure what to do, take a step back from the problem and ask yourself: will graphing this problem out take a long time? If the answer is yes, then there is very likely a much faster, no-graph approach.

The post GRE Coordinate Geometry Shortcut: No Graphs! appeared first on Magoosh GRE Blog.

]]>The post Will there be Parabolas on the GRE? appeared first on Magoosh GRE Blog.

]]>Probably not. Meaning that parabolas can show up on the GRE, but probably won’t show up on the GRE you are taking. That is my initial take based on forums and students experiences. Still, parabolas, along with absolute value graphs, are included in the Practicing to Take the Revised GRE book. ETS obviously didn’t put them there to kill more trees. So, like compound interest and the quadratic equation, parabolas may show up.

What this means for your study plans is that you should only worry about parabolas if you are looking to score above 90%, and have already brushed up on the other concepts on the new GRE math. Meaning, if you are still struggling with number properties or circles, focus there.

That said, below are some important things you have to know about parabolas.

If you draw a line dividing a parabola in the middle, the parabola will be split into two equal halves. This line is known as the axis of symmetry. The shape of the parabola to the right of the axis symmetry is identical to the shape of the parabola the left of it.

The axis of symmetry will either be a vertical line or a horizontal line, but the effect will always be the same: to split the parabola into two equal halves.

The equation for a parabola will always contain a coefficient, meaning that x is always squared: . This may not be too helpful, so just think of it this way: , when graphed, is a parabola. On the other hand, when the x is not squared, say, , we have a simple line.

The equation for a parabola can also be written as . (This would turn out to be if written in the format in the preceding paragraph).

This form is also known as the vertex form and is expressed as . The reason this is call vertex form is it makes finding the vertex very easy.

So how does all of this pertain to the new GRE? Well, remember the line of symmetry? To find it, we simply look at the value of h. In the example above, we have , therefore h is equal to 1.

Finally, there is the vertex, which is either the highest or lowest point depending on the equation (see horizontal vs. vertical, upwards vs. downwards). The vertex can always be represented by . Since h is already 1, we need to find k. This becomes straightforward if we put the following two equations next to each other:

and

Notice that the k is in the place of the 1 on the equation to the left. Therefore . which is also 1. Hence the vertex.

The point at which a straight line intersects a parabola can be found by setting the equation for the line and the equation for the parabola equal to each other. If a line intersects a parabola set the two lines equal to each other.

For instance the line, and the parabola , both equal y so we can set them to equal to each other: . We get , which equals . Therefore .

To find y, we simply plug into either equation (they will always be the same). The answer then is 1. So the line and the parabola intersect at .

Sometimes a question will simply ask you at how many points a line intersects a parabola. A line can intersect a parabola at zero points, one point, or two points.

Sometimes simply graphing out the parabola and the line is the easiest way to answer such a question.

Let’s have a look at two different parabolas:

The two equations are identical, save that the x and y have been swapped. Whenever a parabola is equal to x it is horizontal, meaning that the axis of symmetry is horizontal.

When the parabola is equal to y, the axis of symmetry is vertical.

In the case of the parabola opens to the right. In the case of the parabola opens upward.

Now have a look at the following equations:

This pair of equations is identical to the ones above, except for there is a negative in front of both and . This negative flips the direction of each parabola.

opens to the left and opens downwards.

If this all seems very abstract take a look at the video above.

Finally, a parabola can change its shape depending on whether the coefficient (or number) next to is less than or greater than one. For instance, in the equation

the coefficient is 3, so we will have a ‘skinny’ parabola—that is, the great the coefficient the skinnier the parabola. On the other hand, in , the coefficient is so we will have a ‘fat’ parabola.

(Skinny and fat are my terms not standard math-ese. So if you whip them out in front of math inclined folk you are likely to get some very quizzical looks).

There is much to learn on parabolas, as you can see from the post above. However, there is little likelihood that a parabola will even show up on the test so study only if you are aiming for a top math score.

The accompanying video should hopefully make what I covered in this post far less abstract.

The post Will there be Parabolas on the GRE? appeared first on Magoosh GRE Blog.

]]>The post GRE Math Formulas: How to (Not) Use Them appeared first on Magoosh GRE Blog.

]]>I’m not actually a fan of formulas. Instead, I encourage students to think critically about how formulas are derived. That way, these students are able to have a stronger intuitive sense of the way the math *behind* the formula works.

For instance, say you have a 30-60-90 triangle. Many students falter because they always mix up the sides, especially the side that takes the radical sign. Is it a , or a ?

To think of the proportions intuitively, simply remember that, in a 30-60-90 triangle, the shortest side is always half the length of the longest side. Therefore, we have a ,and a *. * The middle side will have to be less than , so it will either be , or .

As to which one, remember that a 30-60-90 triangle is a right triangle. So, if we use the Pythagorean theorem (which you should definitely be able to execute quickly and accurately), then the shortest side squared (which is 1) subtracted from the longest side squared is equal to 3 (in this case, I just assumed *x* is equal to 1 so that the shortest side and longest side are 1 and 2, respectively). Using the Pythagorean theorem, we square this number, and get a .

If you’ve followed me this far, now you have a way for testing the sides of a 30-60-90, instead of relying on a formula (which can be stressful because you may always forget it…just don’t forget the Pythagorean theorem).

Relying on formulas can also give us formula blindness. That is, even though we’ve remembered a formula, we try to apply it to a problem even when the problem is asking for something different. The reason students often fall into this trap is because a question may use language that is similar to the language you’d expect to conform to the formula.

Let’s take the following problem:

*One side of a right triangle **is twice the length of another side.*

Column A | Column B |
---|---|

The degree measure of the smallest angle of triangle | 30 |

- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given

Your first inclination is to think, oh, it’s a 30-60-90 triangle. After all, I just told you that in a 30-60-90 triangle, the longest side is always twice the shortest. However, the GRE is always trying to trap you, especially on Quantitative Comparison. To avoid becoming a victim of their trap, you must think through the problem, instead of blindly relying on your formula.

With this problem, the twist is that the leg that is twice as long as the shortest leg could also be the middle leg. In that case, we would have a triangle that is not a 30-60-90 triangle. So, the shortest angle would be something different from 30 degrees (it is actually less than 30 degrees). Because Quantitative Comparison does not require you to know the exact number of that angle (something that would be beyond the scope of the GRE), the answer is (D).

Had the only possible scenario been a 30-60-90 triangle, then the answer would be (C). By thinking through the problem, and not simply relying on the formula, we were able to find a different possible triangle that conforms to the information in the problem (*one side of a right triangle…)*.

That is not to say one should be able to intuit every formula. Doing so would only make things more complex. And, for the GRE quantitative section, we want, whenever we can, to make things easier for ourselves.

So, below are a few formulas, mostly from geometry, that you should memorize, especially if you are looking to score in the 700s.

**Area of an equilateral triangle:**

(where s is a side of the equilateral triangle)

**Volume of a sphere:**

**Combined Work Rate:**

Once you’ve memorized these formulas, you should practice them on relevant problems so that applying the formulas becomes natural. You should also be aware when the formula doesn’t completely apply, such as in the case above. Or, when you can find a way outside of the formula to solve the problem, i.e. just because a problem deals with a sphere does not mean you will have to rely on the formula above.

*Takeaways*

*Ultimately, the GRE is testing the way you think. And simply plugging in a bunch of values to a set formula doesn’t test thinking skills insomuch as it tests your ability to memorize a formula. And, trying to memorize a formula is often more difficult than knowing how that formula was derived.*

*In upcoming posts, I’m going to go through and deconstruct formulas that I feel are laborious. Instead, I am going to help you understand how a formula was derived, and then show you some shortcuts, so you are not spending extra time plugging and chugging through some cumbersome formula.*

The post GRE Math Formulas: How to (Not) Use Them appeared first on Magoosh GRE Blog.

]]>