November 17, 2018 November 17, 2018 07:00 AM PST 09:00 AM PST Nov. 17, 7 AM PST. Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT. November 17, 2018 November 17, 2018 09:00 AM PST 11:00 AM PST Join the Quiz Saturday November 17th, 9 AM PST. The Quiz will last approximately 2 hours. Make sure you are on time or you will be at a disadvantage.
Author 
Message 
TAGS:

Hide Tags

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 473

Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 05:09
Question Stats:
35% (03:03) correct 65% (02:47) wrong based on 49 sessions
HideShow timer Statistics
[GMATH course practice question] Points \((x,y)\) in the rectangular coordinate plane such that \(\,y = {x^2} + mx + \left( {8  m} \right)\,\) are presented in the graph shown, where \(m\) is constant. What is the value of \(k+p\) ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Official Answer and Stats are available only to registered users. Register/ Login.
Attachments
03Set18_7m.gif [ 6.22 KiB  Viewed 765 times ]
_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12questions Mock included) to gain a 50% discount!



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 473

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 08:19
fskilnik wrote: [GMATH course practice question]
Points \((x,y)\) in the rectangular coordinate plane such that \(\,y = {x^2} + mx + \left( {8  m} \right)\,\) are presented in the graph shown, where \(m\) is constant. What is the value of \(k+p\) ?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 I am sorry no one contributed (yet), but I hope you all enjoy my solution! \(? = k + p\) \(y = {x^2} + mx + \left( {8  m} \right)\,\,\,\,\,\,,\,\,\,m\,\,{\text{cte}}\,\,\,\,\,\left( * \right)\) \(k = {x_{{\text{vertex}}}}\mathop = \limits^{\left( \odot \right)} \,\,  \frac{m}{{2 \cdot 1}}\,\,\,\,\, \Rightarrow \,\,\,\,\,m =  2k\,\,\,\left( {**} \right)\) \(\left( {k,0} \right)\,\,\, \in \,\,\,{\text{graph}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,0 = \,\,{k^2} + mk + \left( {8  m} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\,0 =  {k^2} + 2k + 8\) \(0 =  {k^2} + 2k + 8\,\,\,\,\mathop \Rightarrow \limits^{{\text{Sum}}\, = \,2\,\,,\,\,\,{\text{Product}}\, = \,\,  8} \,\,\,\,k =  2\,\,\,{\text{or}}\,\,\,k = 4\,\,\,\mathop \Rightarrow \limits^{k\, < \,\,0\,\,\,\left( {f{\text{igure}}} \right)} \,\,\,\,k =  2\) \(\left( {0,p} \right)\,\,\, \in \,\,\,{\text{graph}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,p = \,\,{0^2} + m \cdot 0 + \left( {8  m} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\,p = 8 + 2k = 8 + 2\left( {  2} \right) = 4\) \(? = k + p =  2 + 4 = \boxed2\) Reminder: \(\left( \odot \right)\,\,y = a{x^2} + bx + c\,\,\,\,\left( {a \ne 0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\,{x_{{\text{vertex}}}} =  \frac{b}{{2a}}\) The above follows the notations and rationale taught in the GMATH method. Regards, fskilnik.
Attachments
03Set18_7m.gif [ 6.22 KiB  Viewed 713 times ]
_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12questions Mock included) to gain a 50% discount!



Intern
Joined: 30 May 2017
Posts: 11

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 10:14
let's say that m = 4
y = sqr(x)+4x+2 =sqr (x+2)
when y = 0 x=2 (k)
when x=0 y =4 (p)
k+p =42 = 2 Anwser B



Intern
Joined: 30 May 2017
Posts: 11

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 10:15
let's say that m = 4
y = sqr(x)+4x+2 =sqr (x+2)
when y = 0 x=2 (k)
when x=0 y =4 (p)
k+p =42 = 2 Anwser B



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 473

Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 10:57
amascarenhas wrote: let's say that m = 4
y = sqr(x)+4x+2 =sqr (x+2)
when y = 0 x=2 (k)
when x=0 y =4 (p)
k+p =42 = 2 Anwser B Hi, amascarenhas! I am sure you have meant x^2 in the place of sqr(x) and (x+2)^2 in the place of sqr(x+2) ... The fact that m = 4 makes the expression y = (x+2)^2 DOES guarantee that the x_vertex of the parabola is 2 (=k), exactly the opposite of half the value of this value of m, what we know (a posteriori) is the correct relationship between these two. More than that, y_vertex will be zero (for x=2), exactly as the figure suggests. (The vertex of the parabola must be on the xaxis.) Thank you for your contribution! Regards, fskilnik. P.S.: let me be more clear. If you had chosen (say) m=2, you would find the x_vertex = 1 (negative, fine) but the y_vertex would be 5, therefore the vertex would be point (1,5) , that violates the question stem (figure). In other words, m=4 is not "free" as we wished, therefore exploring "any" particular case would not be possible here.
_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12questions Mock included) to gain a 50% discount!



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 473

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 11:29
Let me present a second possible solution for the interested readers: \(? = k + p\) \(y = {x^2} + mx + \left( {8  m} \right)\,\,\,\,\,\,,\,\,\,m\,\,{\text{cte}}\,\,\,\,\,\left( * \right)\) \(0\mathop = \limits^{{\text{single}}\,\,{\text{root}}} \Delta = {m^2}  4\left( {8  m} \right)\,\,\,\,\, \Rightarrow \,\,\,\,{m^2} + 4m  32 = 0\,\,\,\mathop \Rightarrow \limits^{{\text{Sum}}\, = \,  4\,\,,\,\,\,{\text{Product}}\, = \,\,  32} \,\,\,\,m =  8\,\,\,{\text{or}}\,\,\,m = 4\) \(0\mathop > \limits^{{\text{figure}}} k = {x_{{\text{vertex}}}}\mathop = \limits^{\left( \odot \right)} \,\,  \frac{m}{{2 \cdot 1}}\,\,\,\,\, \Rightarrow \,\,\,\,\,m > 0\,\,\,\,\, \Rightarrow \,\,\,\,\,m = 4\,\,\,\,\,\mathop \Rightarrow \limits^{k =  m/2} \,\,\,\,\,k =  2\) \(\left. \begin{gathered} \left( {0,p} \right)\,\,\, \in \,\,\,{\text{graph}} \hfill \\ m = 4 \hfill \\ \end{gathered} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,p = \,\,{0^2} + 4 \cdot 0 + \left( {8  4} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,p = 4\) \(? = k + p =  2 + 4 = \boxed2\) Reminder: \(\left( \odot \right)\,\,y = a{x^2} + bx + c\,\,\,\,\left( {a \ne 0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\,{x_{{\text{vertex}}}} =  \frac{b}{{2a}}\) The above follows the notations and rationale taught in the GMATH method. Regards, fskilnik.
_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12questions Mock included) to gain a 50% discount!



Senior Manager
Joined: 04 Aug 2010
Posts: 305
Schools: Dartmouth College

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
03 Sep 2018, 17:18
fskilnik wrote: [GMATH course practice question]
Points \((x,y)\) in the rectangular coordinate plane such that \(\,y = {x^2} + mx + \left( {8  m} \right)\,\) are presented in the graph shown, where \(m\) is constant. What is the value of \(k+p\) ?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Since the graph intercepts the xaxis to the left of the yaxis and has only one solution, its graph is as follows: y = (xk)², where k<0. The equation above has an xintercept at (k, 0), as shown in the figure above. The answer choices represent the sum of k (the xintercept) and p (the yintercept). Since the answer choices are very small, test negative values for k that are close to 0. Case 1: k=1 Plugging k=1 into y = (xk)², we get: y = (x(1))² = (x+1)² = x² + 2x + 1. Since the prompt indicates that y = x² + mx + (8m), Case 1 implies that m=2 and that 8m = 1. The equations in red contradict each other, implying that Case 1 is not viable. Case 2: k=2 Plugging k=2 into y = (xk)², we get: y = (x(2))² = (x+2)² = x² + 4x + 4. Since the prompt indicates that y = x² + mx + (8m), Case 2 implies that m=4 and that 8m = 4. The equations in green are both viable for m=4, implying that Case 2 is correct and that the equation of the graph is y = (x+2)². Since y=4 when x=0, the value of p = 4. Thus, k+p = 2 + 4 = 2.
_________________
GMAT and GRE Tutor Over 1800 followers Click here to learn more GMATGuruNY@gmail.com New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and longdistance. For more information, please email me at GMATGuruNY@gmail.com.



Intern
Joined: 22 Jun 2018
Posts: 22
Location: India
GPA: 4
WE: Research (Telecommunications)

Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
Updated on: 05 Sep 2018, 19:10
Hi,
When we look at the equation y=x^2+mx+(8−m) and at the graph we see two things:
1) The equation has only one root. 2) At x=0 , it has a positive intercept i.e. (8m) > 0
Now When we have only one root ( which are also equal and real ) then according to quadratic equations the determinant shall be 0 (B^24AC=0) so solve for m^2  4(8m) = 0
We shall get two values 8, 4 but these should support the 2nd point above. Hence we conclude that m=4. On placing the value of m in the equation we get x^2 + 4x + 4 which is (x +2 )^2 and hence x=2
k is the value of x when y=0 so in other terms k is the root of the equation hence k=2 p is the y intercept when x=0 i.e. p = ( 8 m ) = 4 Hence 4+(2) = 2 (Answer)
Originally posted by anse on 05 Sep 2018, 18:32.
Last edited by anse on 05 Sep 2018, 19:10, edited 1 time in total.



Manager
Joined: 04 Oct 2017
Posts: 67

Re: Points (x,y) in the rectangular coordinate plane such that
[#permalink]
Show Tags
05 Sep 2018, 18:59
amascarenhas wrote: let's say that m = 4
y = sqr(x)+4x+2 =sqr (x+2)
when y = 0 x=2 (k)
when x=0 y =4 (p)
k+p =42 = 2 Anwser B Hi, Thank you for the explanation. How can we assume a value for m?????




Re: Points (x,y) in the rectangular coordinate plane such that &nbs
[#permalink]
05 Sep 2018, 18:59






