Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 29 Jul 2009
Posts: 107

If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
Updated on: 26 May 2015, 05:36
4
This post received KUDOS
29
This post was BOOKMARKED
Question Stats:
49% (02:24) correct 51% (02:11) wrong based on 442 sessions
HideShow timer Statistics
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
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by apoorvasrivastva on 02 Mar 2010, 08:10.
Last edited by Bunuel on 26 May 2015, 05:36, edited 4 times in total.
Edited the OA.



Intern
Joined: 22 Nov 2009
Posts: 29

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
02 Mar 2010, 10:54
2
This post received KUDOS
3
This post was BOOKMARKED
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!)
_________________
kudos +1 ?



Math Expert
Joined: 02 Sep 2009
Posts: 44588

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
02 Mar 2010, 18:32
7
This post received KUDOS
Expert's post
14
This post was BOOKMARKED
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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 29 Jul 2009
Posts: 107

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
03 Mar 2010, 00:01
@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



Math Expert
Joined: 02 Sep 2009
Posts: 44588

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
03 Mar 2010, 00:18
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 \(f(1)>f(2)\).
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 04 Mar 2010
Posts: 2

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
04 Mar 2010, 21:22
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.



Manager
Joined: 03 May 2013
Posts: 72

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
25 May 2015, 18:22
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



Math Expert
Joined: 02 Sep 2009
Posts: 44588

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
26 May 2015, 05:51
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 4455 times ]
Also check parabola chapter HERE. Hope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 02 Jul 2015
Posts: 106

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
08 Oct 2015, 06:35
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!



Manager
Status: Turning my handicaps into assets
Joined: 09 Apr 2017
Posts: 77

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
07 Jan 2018, 09:46
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.
_________________
If time was on my side, I'd still have none to waste......



Math Expert
Joined: 02 Sep 2009
Posts: 44588

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
07 Jan 2018, 09:49
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,
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Status: Turning my handicaps into assets
Joined: 09 Apr 2017
Posts: 77

Re: If the parabola represented by f(x) = ax^2 + bx + c passes [#permalink]
Show Tags
07 Jan 2018, 10:25
I got it. We can know it when putting all the 3 points on a plane. I thought there's another special way. Thanks
_________________
If time was on my side, I'd still have none to waste......




Re: If the parabola represented by f(x) = ax^2 + bx + c passes
[#permalink]
07 Jan 2018, 10:25






