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In the xy-plane shown, the shaded region consists of all points that l

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In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

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Last edited by Bunuel on 24 Jan 2016, 02:18, edited 1 time in total.
Edited the question and added the image.
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 27 Jan 2016, 17:01
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dgboy765 wrote:
Image
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

Dear dgboy765,
I'm happy to help. :-)

Here are the basic ideas. Consider the graphs of the form y = [some expression of x]. These can include
a) oblique lines, y = mx + b (e.g. y = (3/7)x + (5/7))
b) parabolae: e.g. y = x^2, or y = x^2 - 4
c) higher powers of x: e.g. y = x^5 - x^3
d) square roots: y = sqrt(x - 3)
e) all kinds of other exotic curves: y = 2^x, or y = (1 + x)/(1 - x), or etc.
For the purpose of this discussion, the nature of that variety doesn't matter. What I am going to say applies to all graphs of the form y = [expression of x]. I will use the parabola y = x^2 - 4 as my example graph, but what I am saying about this graph applies to any graph of the form y = [expression of x]

The first may be obvious: all the values (x, y) that satisfy the equation y = x^2 - 4 must lie exactly on the graph of y = x^2 - 4. More generally, the graph of any equation is the set of all ordered pairs that satisfy this equation.

Now, think about any point on that line. Imagine we take an (x, y) that live on the graph, and we change the y-coordinate to make it bigger. That would result in a new point that is not on the graph but above the graph. This corresponds to the inequality y > x^2 - 4. Any ordered pair that satisfies this inequality is somewhere above the graph.

Similarly, any ordered pair that satisfies the inequality y < x^2 - 4 lies somewhere below the graph.

In this question, notice that for point (a, b), a is the x-coordinate and b is the y-coordinate. The shaded region shown in the picture is below the x-axis, so y < 0, and it is above the parabola y = x^2 - 4, so it is entirely defined in terms of these two inequalities:
y < 0 and y > x^2 - 4
The prompt already specified that b < 0, and that takes care of the first inequality. Notice that statement #2 is just the second inequality written in terms of a & b instead of x & y. That's why it's sufficient. Once we specify both inequalities, we have specified the shaded region entirely.

On a side note, notice that y = x, the line through the origin with a slope of 1, is the line that contains all points with equal x- and y-coordinates. The inequality y > x are all the points above y = x, and the inequality x > y are all the points below y = x.

Does all this make sense?
Mike :-)
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In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 27 Jan 2016, 20:24
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dgboy765 wrote:
Image
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

[Reveal] Spoiler:
Attachment:
2016-01-24_1416.png


Hi,
In very simple terms to solve this Q..

The type of parabola etc is amplified by mike in the above post...
A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of \(x^2\)..
here it is positive, so the parabola will be open upwards and will have a mininmum value at\(x=\frac{-b}{2a}\) or 4/2=2.. and the value is \(2^2-4*2=-4\)...

lets see the statements now..

(1) \(0 < a < 4\)..
nothing about b....
so point (a,b) can be anywhere depending on value of b
at a=3.99, b can be -4, so will be outside the graph or at some point inside..
and at a=2, b can be -3.99, it will be inside the graph..
so insuff..

(2) \(a^2 - 4a < b\)
the moment you see this equation, its similarity with the original equation y=x^2 - 4x should strike you..
we substitute a and b as x and y in the eq we get b=a^2-4a...
since the equation \(b=a^2-4a\) is that of the the line..
a^2-4a< b will be inside the parabola and a^2-4a>b will be outside it...
so suff..
you can test this with, say at the x axis..
at a=4, b=0..
\(a^2-4a=b... 4^2-4*4=b=0\)..
so if \(a^2-4a<0, 0<a<4\) satisfies the condition for within the shaded portion and so suff..
the moment a^2-4a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..
hope it helped you in some way
_________________

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Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 14 Sep 2016, 06:47
Hi Mike
Actually, I can not get the concept behind your explanation.
Please explain more taking into account the basic concepts since I did not master the Coordinate concepts.
Thank you in advance.
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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hatemnag wrote:
Hi Mike
Actually, I can not get the concept behind your explanation.
Please explain more taking into account the basic concepts since I did not master the Coordinate concepts.
Thank you in advance.

Dear hatemnag,
I'm happy to respond. :-)

My friend, before I answer your question, I am going to challenge you. I am going to challenge you to ask a better, more thorough question. See this blog article:
Asking Excellent Questions

You see, the question you asked was very vague and general. Vague questions are poor questions. It's perfectly fine that you don't understand some basic concepts about Coordinate Geometry and that you want to learn more--in fact, it's wonderful that you are asking for help! The trouble is, I have absolutely no idea what you already know and what you need. Your vague question completely leaves me in the dark.

An excellent question would involve making explicitly clear, exactly and specifically, what parts you understand and what parts you don't understand. For example, you could go through each point I make in my Jan. 27, 2016 post above, and tell me exactly what you understand and exactly what confuses you about each item.

Of course, a more specific, more detailed question, will help me respond to you more effectively, but what students often fail to understand: I recommend crafting an excellent question, not for me, but for you! The process of putting all the effort into writing an excellent question will be tremendously helpful to you: by explaining all this, you will force yourself to make connections and have realizations, and all this effort will prime your mind so that my more detailed response will be that much more helpful for you. It's much harder to produce an excellent question, and, in fact, all the effort it takes to craft an excellent question is actually an investment in your own understanding and learning. When a student poses an excellent question to the teacher, it's a total win-win scenario. This is precisely why asking excellent questions is one of the habits of excellence.

Therefore, my friend, I am going to challenge you to write an excellent question. Craft the highest quality question you can, making explicitly and specifically clear precisely which parts you understand and don't understand about coordinate geometry. It's fine to have a ton of questions, that's great, but it's important for both of us to appreciate the basic ideas that you do understand, because all learning is based on what you already grasp.

Does all this make sense?

Mike
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 12 Feb 2017, 21:13
hi there, mikemcgarry and chetan2u can someone help me out? I am taking the GMAT in 8 days (my practice CATs have been in the 700 range Q49, V40), but this question came as a bit of a shock to me. Note, I have not read either of your explanations yet because this concept looks new to me. What is "a" and "b" referring to? Is it the a and b within the quadratic equation (ax^2+bx+c=0)? For the most part, the only questions I have seen related to graphs in my practice is lines (y=mx+b). I will read your explanations shortly, but first, is there a gap in my understanding of the basics? I think I remember from grade school this U shape being a "Parabola." Is that correct? If so, what about Parabola's do we need to know for the GMAT?
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 13 Feb 2017, 04:08
gzimmer wrote:
hi there, mikemcgarry and chetan2u can someone help me out? I am taking the GMAT in 8 days (my practice CATs have been in the 700 range Q49, V40), but this question came as a bit of a shock to me. Note, I have not read either of your explanations yet because this concept looks new to me. What is "a" and "b" referring to? Is it the a and b within the quadratic equation (ax^2+bx+c=0)? For the most part, the only questions I have seen related to graphs in my practice is lines (y=mx+b). I will read your explanations shortly, but first, is there a gap in my understanding of the basics? I think I remember from grade school this U shape being a "Parabola." Is that correct? If so, what about Parabola's do we need to know for the GMAT?


Hi gzimmer,

a and b refer to a point in the plane where a is on x-axis and B on y-axis..
You are correct that such equation refers to parabola.
Also parabola if asked would be basic, as for instance one can see the similarities in two equations- one in Q and other in statement II.
And that would also mean you are likely to be doing well in Quant till then.

Now with just 8 days left, just do the basic and do not spend too much time on topics rarely seen ..
All the best.
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 13 Feb 2017, 15:16
dgboy765 wrote:
Image
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

[Reveal] Spoiler:
Attachment:
2016-01-24_1416.png


Hi,
I solved it this way:

remember that stem says b<0.

st1. clearly insufficient because say y= -5 then it is not in shade region but for y greater that -4 it is in region.

st2. you can rewrite it as a^2-4a+4<b+4 (add 4 to both sides) then LHS is (a-2)^2 and RHS is b+4 ---> (a-2)^2<a+4 as LHS is something always positive then it can not be smaller than something negative thus b+4 must be greater than 0, therefore we have -4<b<0 hence 0<b+4<4. ---> (a-2)^2<4 ---> |x-2|<2 --> -2<x-2<2 --> 0<x<4. it is in the region. suff.
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 06 Mar 2017, 10:39
i am not able to understand the solution
can anyone please help
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 05 Jun 2017, 06:43
chetan2u wrote:
dgboy765 wrote:
Image
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

[Reveal] Spoiler:
Attachment:
2016-01-24_1416.png


Hi,
In very simple terms to solve this Q..

The type of parabola etc is amplified by mike in the above post...
A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of x^2..
here it is positive, so the parabola will be open upwards and will have a mininmum value at x=-b/2a or 4/2=2.. and teh value is 2^2-4*2=-4...

lets see the statements now..

(1) 0 < a < 4..
nothing about b..
at a=3.99, b can be -4, so will be outside the graph ..
and at a=2, b can be -3.99, it will be inside the graph..
so insuff..

(2) a^2 - 4a < b
the moment you see this equation, its similarity with the original equation y=x^2 - 4x should strike you..
we substitute a and b as x and y in the eq we get b=a^2-4a...
since the equation b=a^2-4a is that of the the line..
a^2-4a< b will be inside the parabola and a^2-4a>b will be outside it...
so suff..
you can test this with, say at the x axis..
at a=4, b=0..
a^2-4a=b... 4^2-4*4=b=0..
so if a^2-4a<0, 0<a<4 satisfies the conditi0n for within the shaded portion and so suff..
the moment a^2-4a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..
hope it helped you in some way



In Statement 1 couldn't Point B be anything? For instance -100000 ? little bit confused why you picked -3.99 and and -4 for b ? or am I missing something ?


Im sure Im missing something or the questions is just very basic

I mean from statement 1: no information about b at all --> so clearly insufficient.

Statement 2: it explicitly tells us that a,b lies above the graph and it is given that b<0
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 18 Aug 2017, 17:56
Hi VeritasPrepKarishma mikemcgarry shashankism Bunuel

Can you suggest flaw in below approach:

Given: Question has asked me whether a point (a,b) lies on shaded curve.
I hope there is no discrepancy between whether the shaded curve includes the parabola.
As per mine understanding it does not.

Essentially the parabola is in quadrant IV and hence (x,-y) should be co-ordinates of points in this quadrant.

St 1: 0<a<4 -> I simply took in random values between 0 to 3.9 and found that since x is positive st 1 is sufficient.
After reading the solutions and above explanation found that for two different values of a/x, say 1 or 8 I am not alway
getting positive values of y hence st 1 is insufficient.

St 2: at first glance the equation translates to x^2 - 4x < y since (a,b) have to in shaded region but for these how did we
intuitively assume that (a,b) is on curve? this query is especially pertaining to above explanation by mikemcgarry

As per my understanding it seems we have put an upper gap on parabola by saying b<0 in question stem. Any view on this?
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Re: In the xy-plane shown, the shaded region consists of all points that l [#permalink]

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New post 20 Aug 2017, 01:19
daviddaviddavid wrote:
chetan2u wrote:
dgboy765 wrote:
Image
In the xy-plane shown, the shaded region consists of all points that lie above the graph of y=x^2 - 4x and below the the x-axis. Does the point (a,b) (not shown) lie in the shaded region if b<0?

(1) 0 < a < 4
(2) a^2 - 4a < b


Source: Official GMAT Quantitative Review 2016
P. 162 DS #124

Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.)

[Reveal] Spoiler:
Attachment:
2016-01-24_1416.png


Hi,
In very simple terms to solve this Q..

The type of parabola etc is amplified by mike in the above post...
A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of x^2..
here it is positive, so the parabola will be open upwards and will have a mininmum value at x=-b/2a or 4/2=2.. and teh value is 2^2-4*2=-4...

lets see the statements now..

(1) 0 < a < 4..
nothing about b..
at a=3.99, b can be -4, so will be outside the graph ..
and at a=2, b can be -3.99, it will be inside the graph..
so insuff..

(2) a^2 - 4a < b
the moment you see this equation, its similarity with the original equation y=x^2 - 4x should strike you..
we substitute a and b as x and y in the eq we get b=a^2-4a...
since the equation b=a^2-4a is that of the the line..
a^2-4a< b will be inside the parabola and a^2-4a>b will be outside it...
so suff..
you can test this with, say at the x axis..
at a=4, b=0..
a^2-4a=b... 4^2-4*4=b=0..
so if a^2-4a<0, 0<a<4 satisfies the conditi0n for within the shaded portion and so suff..
the moment a^2-4a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..
hope it helped you in some way



In Statement 1 couldn't Point B be anything? For instance -100000 ? little bit confused why you picked -3.99 and and -4 for b ? or am I missing something ?


Im sure Im missing something or the questions is just very basic

I mean from statement 1: no information about b at all --> so clearly insufficient.

Statement 2: it explicitly tells us that a,b lies above the graph and it is given that b<0



hi..

since b is not given and no relation exists between a and b, b can take any value..
you are correct in your understanding...two points given were just to illustrate that b could be inside or outside graph
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html


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Re: In the xy-plane shown, the shaded region consists of all points that l   [#permalink] 20 Aug 2017, 01:19
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