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Posts: 44
Updated on: 26 May 2015, 04:36
Originally posted by apoorvasrivastva on 02 Mar 2010, 07:10.
Last edited by Bunuel on 26 May 2015, 04:36, edited 4 times in total.
Last edited by Bunuel on 26 May 2015, 04:36, edited 4 times in total.
Edited the OA.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
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Question Stats:
49% (02:54) correct
51%
(02:44)
wrong
based on 398
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If the parabola represented by f(x) = ax^2 + bx + c passes through points (-3,0) , (0,3) and (5,0), which of the following must be true?
I. f(-1) > f(2)
II. f(1) > f(0)
III. f(2) > f(1)
A. Only I
B. Only II
C. Only III
D. I and II
E. I and III
I. f(-1) > f(2)
II. f(1) > f(0)
III. f(2) > f(1)
A. Only I
B. Only II
C. Only III
D. I and II
E. I and III
apoorvasrivastva
Answer to this question cannot be D, it should be B (II only).
To solve this question you don't need to calculate a, b, and c. We should put the given three points on the XY-plane and we'll get: parabola \(f(x)=ax^2+bx+c\) will have the vertex at \(x=1\), halfway between \(x=-3\) and \(x=5\), as \(f(-3)=0=f(5)\). Plus, as \(f(0)=3\) the parabola would be downward. As the parabola is downward, value of f(x) at x=1, f(1), is the the greatest value of f(x). As we move from x=1 to either of direction the value of f(x) will decrease. So any \(f(m)\) would be greater than \(f(n)\) if \(m\) is closer to \(x=1\) than \(n\). For example \(f(9)>f(10)>f(-12)\) or \(f(6)=f(-4)\) or \(f(-5)=f(7)>f(-100)\).
Hence
I. \(f(-1)>f(2)\) is false, as \(x=2\) is 1 farther from \(x=1\) and \(x=-1\) is 2 farther than \(x=1\). (True inequality would be \(f(-1)<f(2)\)).
II. \(f(1)>f(0)\) is true as \(f(1)\) is vertex and more than any \(f(x)\).
III. \(f(2)>f(1)\) is false as no \(f(x)\) is more than \(f(1)\). (The correct would be \(f(2)<f(1)\)).
Hope it's clear.
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apoorvasrivastva
f(x)=y
Substituting (x,y) as (-3,0) , (0,3) and (5,0), we get the following equations:
0 = 9a-3b+c
3 = c
0 = 25a+5b+c
Solving: a=-1/5, b=2/5, c=3
f(x) = -x^2/5 + x/5 +3
The options are on f(-1), f(0), f(1) and f(2).
f(-1) = -1/5 + 2/5 + 3 = 16/5 = 3.2
f(0) = 3
f(1) = 1/5 + 2/5 + 3 = 18/5 = 3.6
f(2) = -4/5 + 4/5 + 3 = 3
I f(-1) > f(2) true
II f(1) > f(0) true
III f(2) > f(1) false
So, option D, I and II
(not sure if there is an easier way to solve this!)
General Discussion
Posts: 44
@bunnuel i have dited my post for the st I ..it was a typo error i am sorry abt that
it is f(-1) > f(-2)
i am not clear as to how did u get the vertex as x=1
it is f(-1) > f(-2)
i am not clear as to how did u get the vertex as x=1
apoorvasrivastva
Intersection points of parabola with x-axis are \((-3,0)\) and \((5,0)\):
-----(-3)--0----(5)---, as parabola is symmetric, the x coordinate of the vertex must be halfway between \(x=-3\) and \(x=5\) --> \(x=\frac{-3+5}{2}=1\). As parabola is downward, \(f(x)\) naturally will have it's max values at vertex \(x=1\), \(f(1)\).
As for the typo in the stem. If I is saying: \(f(-1) > f(-2)\), then it's true as \(x=-1\) is closer to \(x=1\), than \(x=-2\), which means that \(f(-1)>f(-2)\).
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amazing explanation Bunuel. I would normally have found a, b and c and then solved. Thanks for showing me the way to think more than the way you solved the problem.
Bunuel
HI Bunuel,
Please explain the red part , what i understood from making graph , if we move left from the axis point certainly value of f(x) decreases but if we move right side from the axis where x =1, value of f(x) will increase till certain point and then will decrease, so how can we deduce that value of f(x) is max at x =1
vipulgoel
Check below:
Attachment:
parabola.png [ 11.58 KiB | Viewed 11754 times ]
Also check parabola chapter HERE.
Hope it helps.
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Bunuel
Wonderful approach! thanks a lot!
Bunuel
Hi Bunuel, could you please explain the highlighted part. How do we infer that parabola is downward? According to theory we only know that parabola is downward if a<0.
Mehemmed
Hint: put the given three points on the plane,
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I got it. We can know it when putting all the 3 points on a plane. I thought there's another special way. Thanks
Posts: 1,324
Concept: Once you find the Line of Symmetry of a Downward Opening Parabola, the X Coordinate at this Vertex will give you the HIGHEST VALUE that f(x) = y can have
Rule: to find the Line of Symmetry of a Parabola (in which everything on the Left Side of the Axis/Line is a "MIRROR REFLECTION" of all the Points on the Right Side), find the MID-POINT between the X-Intercepts that cross the X-Axis
between (-3 , 0) and (+5 , 0) ---- the Line of Symmetry will occur at the Vertical Line of: X = +1
Also, since this is a Downward Opening Parabola, the Vertex at f(+1) will also provide the HIGHEST VALUE of y
I. f(-1) > f(2)
f(-1) is +2 Units to the LEFT of the Highest Value at the Vertex f(+1)
However, f(2) is only +1 Unit to the RIGHT of the Highest Value at the Vertex f(+1)
the Value of f(2) will be GREATER than the Value of f(-1)
NOT True
II. f(1) > f(0)
The Highest Value will occur at the Vertex, when X =+1, because this is a Downward Opening Parabola.
Thus f(1) will be greater than any other Value
TRUE
III. f(2) > f(1)
As stated above, the Highest Value will occur at the Vertex when the X-Coordinate is +1 (at the Line of Symmetry) because this is a Downward Opening Parabola.
thus: f(2) < f(1)
III is NOT True
ONLY II must be True
-B-
Rule: to find the Line of Symmetry of a Parabola (in which everything on the Left Side of the Axis/Line is a "MIRROR REFLECTION" of all the Points on the Right Side), find the MID-POINT between the X-Intercepts that cross the X-Axis
between (-3 , 0) and (+5 , 0) ---- the Line of Symmetry will occur at the Vertical Line of: X = +1
Also, since this is a Downward Opening Parabola, the Vertex at f(+1) will also provide the HIGHEST VALUE of y
I. f(-1) > f(2)
f(-1) is +2 Units to the LEFT of the Highest Value at the Vertex f(+1)
However, f(2) is only +1 Unit to the RIGHT of the Highest Value at the Vertex f(+1)
the Value of f(2) will be GREATER than the Value of f(-1)
NOT True
II. f(1) > f(0)
The Highest Value will occur at the Vertex, when X =+1, because this is a Downward Opening Parabola.
Thus f(1) will be greater than any other Value
TRUE
III. f(2) > f(1)
As stated above, the Highest Value will occur at the Vertex when the X-Coordinate is +1 (at the Line of Symmetry) because this is a Downward Opening Parabola.
thus: f(2) < f(1)
III is NOT True
ONLY II must be True
-B-
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