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If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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02 Mar 2010, 07:10
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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
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Last edited by Bunuel on 26 May 2015, 04:36, edited 4 times in total.
Edited the OA.



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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apoorvasrivastva wrote: 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)
f(x)=y Substituting (x,y) as (3,0) , (0,3) and (5,0), we get the following equations: 0 = 9a3b+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!)
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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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02 Mar 2010, 23:01
@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



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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02 Mar 2010, 23:18
apoorvasrivastva wrote: @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 Intersection points of parabola with xaxis 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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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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04 Mar 2010, 20:22
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.



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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25 May 2015, 17:22
Bunuel wrote: apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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. 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



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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26 May 2015, 04:51
vipulgoel wrote: Bunuel wrote: apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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. 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 Check below: Attachment:
parabola.png [ 11.58 KiB  Viewed 4103 times ]
Also check parabola chapter HERE. Hope it helps.
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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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08 Oct 2015, 05:35
Bunuel wrote: apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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. Wonderful approach! thanks a lot!



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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07 Jan 2018, 08:46
Bunuel wrote: apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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. 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.



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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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07 Jan 2018, 08:49
Mehemmed wrote: Bunuel wrote: apoorvasrivastva wrote: 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 please suggest me as to how do i crack this question in 2 mints..i am completely lost on this OA is 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 XYplane 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. 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. Hint: put the given three points on the plane,
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Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
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07 Jan 2018, 09:25
I got it. We can know it when putting all the 3 points on a plane. I thought there's another special way. Thanks




Re: If the parabola represented by f(x) = ax^2 + bx + c passes
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