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In the xy-plane, a parabola intersects with axis-y at point

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In the xy-plane, a parabola intersects with axis-y at point [#permalink] New post 10 Dec 2010, 10:03
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In the xy-plane, a parabola intersects with axis-y at point (0,y). Is y < 0 ?

(1) The vertex of parabola is (2,-5)
(2) The parabola intersects with axis-x at point (-2,0) and (6,0)
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Re: #32 parabola intersts Y axis [#permalink] New post 10 Dec 2010, 12:52
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aalriy wrote:
In the xy-plane, a parabola intersects with axis-y at point (0,y). Is y<0.

I. The vertex of parabola is (2,-5)
II. The parabola intersects with axis-x at point (-2,0) and (6,0)


Though it's possible to solve this question algebraically the easiest way will be to visualize it and draw on a paper.

(1) The vertex of parabola is (2,-5) --> the vertex is in the IV quadrant: if the parabola is downward it'll have negative y-intercept, but if it's upward then it can have positive as well as negative y-intercept. Not sufficient.

(2) The parabola intersects with axis-x at point (-2,0) and (6,0) --> now if the vertex is above x-axis then parabola will have positive y-intercept and if its vertex is below x-axis it'll have negative y-intercept. Not sufficient.

(1)+(2) As from (1) the vertex is below x-axis then from (2) we'll have that parabola must have negative y-intercept. Sufficient.

You can look at the diagram below to see that a parabola passing through the given three points must have negative y-intercept only.
Attachment:
MSP139819db6ebe95a9e2a900005889a632a09g5628.gif
MSP139819db6ebe95a9e2a900005889a632a09g5628.gif [ 3.25 KiB | Viewed 2449 times ]


Answer: C.

Hope it's clear.
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Re: #32 parabola intersts Y axis [#permalink] New post 20 Oct 2011, 21:12
Thanks Bunuel for the explanation,
I was thinking since option B gives us (-2,0) and (6,0) vertices, with this we can assume that the parabola is directed upwards and it's Y intersect will be -ve and hence sufficient, but
after reading your explanation I understand it better :)
1+ Kudos to you.
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Re: In the xy-plane, a parabola intersects with axis-y at point [#permalink] New post 27 Jun 2013, 21:30
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Re: In the xy-plane, a parabola intersects with axis-y at point [#permalink] New post 27 Oct 2013, 23:40
Hi Bunuel

What is wrong in the following approach

1) Insufficient
The vertex of parabola is (2,-5) --> the vertex is in the IV quadrant: if the parabola is downward it'll have negative y-intercept, but if it's upward then it can have positive as well as negative y-intercept. Not sufficient

I was thinking from option 2 we can find the product of the roots and sum of the roots
product of roots = c/a = -12

and sum of roots = -b/a = 4

hence come up with an equation y = x^2-4x-12

and from the question stem we have that it intercepts at 0,y

Putting x = 0 we get y = -12 (negative) and B alone is sufficient .
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Re: In the xy-plane, a parabola intersects with axis-y at point [#permalink] New post 28 Oct 2013, 00:13
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sr2013 wrote:
Hi Bunuel

What is wrong in the following approach

1) Insufficient
The vertex of parabola is (2,-5) --> the vertex is in the IV quadrant: if the parabola is downward it'll have negative y-intercept, but if it's upward then it can have positive as well as negative y-intercept. Not sufficient

I was thinking from option 2 we can find the product of the roots and sum of the roots
product of roots = c/a = -12

and sum of roots = -b/a = 4

hence come up with an equation y = x^2-4x-12

and from the question stem we have that it intercepts at 0,y

Putting x = 0 we get y = -12 (negative) and B alone is sufficient .


How did you get y=x^2-4x-12 from c/a=-12 and -b/a=4? You cannot solve c/a=-12 and -b/a=4 to get unique values of a, b, and c.

For example if a=2, c=-12, and b=-4 you'll get 2x^2-24x-8=0:
Attachment:
MSP2141ga1ih773ii7e3di00002h2000h2fiibia6c.gif
MSP2141ga1ih773ii7e3di00002h2000h2fiibia6c.gif [ 3.51 KiB | Viewed 1004 times ]


If a=-1, c=12, and b=4 you'll get -x^2+4x+12=0:
Attachment:
MSP24971d0ia65iig79i4gg000037eb4cg662i7adih.gif
MSP24971d0ia65iig79i4gg000037eb4cg662i7adih.gif [ 3.44 KiB | Viewed 1005 times ]


Or in other words infinitely many parabolas have x intercepts at -2 and 6. You cannot get unique equation only from that info.
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Re: In the xy-plane, a parabola intersects with axis-y at point   [#permalink] 28 Oct 2013, 00:13
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