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Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x? (1)

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Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x? (1)  [#permalink]

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New post 06 Apr 2019, 22:30
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Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x?

(1) f(x+2) have 2 intersections with axis-x.
(2) f(x)+2 have 2 intersections with axis-x.

A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

VeritasKarishma can you please help ?
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Re: Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x? (1)  [#permalink]

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New post 08 Apr 2019, 16:42
This is quite tricky.
So, we have to deal with a parabola. Let's analyse carefully what the statements tell us:

Statement 1: we know that \(f(x+2)\) have 2 intersections with x-axis. So, for \(x + 2 - 2 = x\), values lie on different places of x-axis, but still are two. Sufficient.
Statement 2: \(f(x) + 2\) means that our parabola is shifted upward by two. If its vertex was located at (0,2), for instance, and \(a < 0\), \(f(x)\) would not meet x-axis at any point. So, insufficient.
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Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x? (1)  [#permalink]

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New post 08 Apr 2019, 22:32
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MBA20 wrote:
Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x?

(1) f(x+2) have 2 intersections with axis-x.
(2) f(x)+2 have 2 intersections with axis-x.

A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

VeritasKarishma can you please help ?



(ax^2+bx+c) graph is a parabola which could look like any of these:

Attachment:
Screenshot 2019-04-09 at 11.00.06.png
Screenshot 2019-04-09 at 11.00.06.png [ 27.27 KiB | Viewed 132 times ]

It may intersect x axis in 2 points/1 point of no point. Where it is placed on the co-ordinate depends on the values of the coefficients.

(1) f(x+2) have 2 intersections with axis-x.

The graph of f(x+2) is 2 units to the left of graph of f(x). The graph is exactly the same except it moves 2 units to the left. For example:

Attachment:
Screenshot 2019-04-09 at 10.57.04.png
Screenshot 2019-04-09 at 10.57.04.png [ 34.4 KiB | Viewed 132 times ]

So if f(x + 2) has 2 points of intersection, so will f(x). It will be the same graph 2 units to the right of graph of f(x +2).
Hence this statement alone is sufficient.

(2) f(x)+2 have 2 intersections with axis-x.
f(x) + 2 will be 2 units above the graph of f(x). It is possible that f(x) has two points of intersection too but it is possible that f(x) + 2 is graph B shown in pic above while f(x) is graph A. Hence f(x) may have no point of intersection or 1 point of intersection.
This alone is not sufficient.

Answer (A)
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Does the graph of f(x)=ax^2+bx+c have 2 intersections with axis-x? (1)   [#permalink] 08 Apr 2019, 22:32
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