Today's question of the day

Test Preparation SAT Preparation
1

<p>y= -2x²+bx+5</p>

<p>In the xy-plane, the graph of the equation above assumes its maximum value at . What is the value of b?</p>

<p>A. –8
B. –4
C. 4
D. 8
E. 10</p>

<p>My question is... are we supposed to know this formula y=a(x-h)²+k?
After studying for SAT math for a year I haven't met with this formula in anywhere or I don't remember... I only saw a question that required it once among like 30 official tests... And I thought I don't need it but here it is again in a Question of the Day.
How often do you meet with this formula?</p>

2

<p>No need to memorize that formula if you haven’t come across it elsewhere.</p>

<p>Why don’t you just plug those equations into your graphing calculator, and see which one reaches a maximum at x=2? Start with the middle (b=4), see that the max occurs left of x=2, then try x=8 and see the max at x=2. That’s your answer.</p>

<p>Here’s another way to look at it (using basic calculus):
Differentiate y with respect to x: y’ = -4x + b. You know that we’re dealing with x=2 and a maximum (y’=0), so 0 = -4*2 + b. Solve for b, and b=8.</p>

<p>Don’t forget that there’s always a variety of ways to solve a given problem.</p>

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<p>I’ve taken about 30 practice tests and I have never, ever seen that formula used. I suppose it’s helpful to know it, but I’ve never had to use it to solve a question.</p>

4

<p>I’m wondering how you got that 2 superscripted.</p>

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<p>WiseGuy, thanks, I’ll have to learn to use my graphing calculator. I see it can save me some time. </p>

<p>Philovitist, I just copied it from another website.</p>

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<p>The graph of the parabola y = ax^2 + bx + c attains a maximum or minimum at x = -b/2a. The parabola is also symmetric about this line, x = -b/2a, also called the line of symmetry. </p>

<p>In this particular problem, y = -2x^2 + bx + c, we are told that the maximum is obtained at x =2, therefore, 2 = -b/[2(-2)], which gives b = 8. </p>

<p>This in my opinion is faster than using a calculator, because you would have to plug all of the answer choices and that is too time consuming. </p>

<p>I would rate this a Level 5 question, and if you are targeting a 750+ on the SAT math, I would recommend understanding the concept behind this problem. I believe I have seen one more similar SAT problem on the tests.</p>

7

<p>I’m just reiterating what SATQuantum said; I’m sorry if it seems like copy-paste, I just really love functions.</p>

<p>This is the fastest method that I can think of at present, so here goes:
Because y= -2x² + bx + 5, it is a parabola (y = ax² + bx + c).
We are given the maximum value (2) which is also the x-value of the parabola’s axis of symmetry. The formula to find the x-value of the axis of symmetry is as follows:</p>

<p>x = -b/2a</p>

<p>When we substitute our given values, the equation becomes:</p>

<p>2 = -b/2(-2)</p>

<p>Then all you have to do is solve for b and voila, the answer is 8 :)</p>

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<p>@SATQuantum --</p>

<p>It’s not a big deal, but I would be curious as to when you saw this on a real SAT. I don’t believe I have ever seen it. So up until now, I have told my SAT students not to worry about it. (And yes, I know that “good” math students “should” know this. But there are a lot of things in that category, and if they are not on the SAT, I don’t want to spend time on them!) So if you can confirm this, I’ll have to rethink that advice…</p>

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<p>Thanks for your answers, guys. To me this is new and unknown so repeated explanations help me to understand better and settle it into my brain.</p>

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<p>@pckeller
It would be hard for me to give you the exact reference out of the 3000 SAT official math questions(if I do stumble on it again I will definitely give you the year and month of the exam), but I believe that the question had to do with the understanding of the concept of the line of symmetry. One had to recognize that the y-coordinate of the points, that are the same distance on both sides of the line of symmetry is identical. </p>

<p>Other than that, the interpretation of the equation of parabola y = a(x-k)^2 + k, in terms of shifts of y=x^2 and its relationship to the vertex (h,k) is another important idea. And of course, the fact that they are testing this particular question in the question of the day, definitely means that this idea is fair game for the SAT. </p>

<p>Second Method(does not require knowing the actual equation of the line of symmetry):</p>

<p>If we know that the parabola attains its maximum value at x =2, then the value of y at x = 1 and x=3 would be identical, these two points are the same distance from the axis of symmetry. One could also choose, x = 0 and x = 4, which actually might be more convenient because of x = 0 as one of the points. </p>

<p>If we replace x with 0 in the quadratic equation y= -2x²+bx+5, we obtain
y = 0 + 0 + 5 = 5, </p>

<p>because at x = 4, the y-coordinate of the parabola is identical, when we replace x =4 in the above equation of the parabola we must have y=5, therefore:</p>

<p>y = -2(4)²+b(4)+5 = 5 </p>

<p>or -32 + 4b + 5 = 5 </p>

<p>or -32 + 4b = 0 </p>

<p>or 4b = 32 </p>

<p>and again we obtain b = 8.</p>

<p>Third Method(Algebraic method by completing the square):</p>

<p>Another way to approach this problem would be to complete the square:
y= -2x²+bx+5</p>

<p>y = -2(x² - bx/2) + 5 </p>

<p>y = -2[ (x - b/4)² - b²/16] + 5 </p>

<p>y = -2(x - b/4)² + b²/8 + 5 </p>

<p>Here the expression (x-b/4)² is always positive(square expression), therefore
-2(x - b/4)² is always negative. The largest value of y will be attained when this expression is zero, which would happen when x - b/4 = 0 or x = b/4, we are given that this happens when, x is 2, which again leads to b = 8. </p>

<p>This approach is longer because we are redoing all the work to show that the parabola y = ax² + bx + c, attains its extreme value at x = -b/2a. </p>

<p>Cheers</p>

11

<p>Yeah, I agree that knowing the symmetry is useful. And recognzing a “shifted parabola” as well. These have both been tested. I’ve just never seen -b/2a tested (yet).</p>

<p>And I wonder about using the Question of the Day as an indicator of future topics. How much do we know about where these questions come from? I always suspected that writing them is the job they give the new guy or whoever lost the office pool that week…</p>

<p>Merry Christmas</p>

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<p>@pckeller </p>

<p>I honestly don’t know the story behind the question of the day(QOTD). Perhaps they are using it to test new types of questions. I personally take in to account any new concept that they are testing in these questions. I wonder if they use QOTD for experimental purposes just like the experimental section on the real exam. But this would mean that questions on QOTD would appear on the real exams in the future. </p>

<p>I know that they do remove the webpages containing old QOTDs, I might see if I can get my hand on old QOTDs and see if any of those repeat on recent administrations.</p>

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<p>No, there have been QOTDs which can require to find the maximum or minimum value using -b/2a, which is obtained when the standard quadratic equation is differentiated w.r.t the variable x.</p>

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<p>It is only a recent phenomena that you cannot go back and view old QOTD. A little over a year ago they changed their URL configuration to redirect requests for old questions to the current day’s question. I have QOTD e-mails going back a while and can state unequivocally that the College Board rotates the same questions through over and over again. Therefore I think you will never see these questions on a real test. </p>

<p>I don’t think these are retired questions from old exams either. It is not unusual to find clunky questions in the QOTD, and even outright typos from time to time. I don’t think these questions are vetted at all and are only slightly more useful than 3rd party practice questions. Same goes for the quizzes in the online course. </p>

<p>As far as the referenced question is concerned, as you point out, knowing the formula for determining the x-value for the axis of symmetry is completely unnecessary. The official explanation is needlessly and misleadingly mathematical. Anyone in a position to understand the explanation probably didn’t need an explanation in the first place. Typical for the College Board.</p>

<p>The vertex of the parabola is given. One only needs to recognize that parabolas are symmetrical and y must =5 when x=4. Plug those in and solve for b. That is all that is needed to answer the question. This is typical in that what is really a logic question is disguised as a math question.</p>

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<p>The answer is no, you don’t need to know it. This is a difficult question. But it is a difficult logic question not a difficult math knowledge question. Admittedly, a solid knowledge of the math can make some of these questions less difficult.</p>

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<p>I think you are probably close to the truth there. I actually have a cynical view of the whole test. The College Board markets the SAT as an achievement test and every piece of public information, including QOTD answer explanations, is constructed to support that marketing position. I sometimes think the QOTD are constructed not to replicate actual test questions but to support the College Board’s narrative of what the test is trying to measure.</p>

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<p>@YZamyatin Thank you for your comments and insight. I had tried earlier to access the past QOTDs and I had also noticed that they had change the url configuration. I went back today and I went through all the questions(math only) and categorized them by topic and difficulty level. I have listed them on the following page on my site:</p>

<p>[satquantum</a> - SAT QOTD](<a href=“http://www.satquantum.com/sat-qotd/]satquantum”>http://www.satquantum.com/sat-qotd/) </p>

<p>I do need to reorganize it a little bit better and also include the video explanations. </p>

<p>You are right that they actually just repeat these questions. I was able to find the quadratic question that was posted in this post in a 2011 question of the day. Here is the link: [The</a> Official SAT Question of the Day](<a href=“SAT Practice and Preparation – SAT Suite | College Board”>SAT Practice and Preparation – SAT Suite | College Board)</p>

<p>After I had exhausted the 130 or so math questions in the archive they just started to repeat, and I am presuming they have been doing this over the years, especially if you have several years old emails. </p>

<p>As for the source of these questions, there were several that are from old official tests, or slight variations. Here is one example:
[The</a> Official SAT Question of the Day](<a href=“SAT Practice and Preparation – SAT Suite | College Board”>SAT Practice and Preparation – SAT Suite | College Board)</p>

<p>I believe there is a similar question in the SAT official guide.</p>

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<p>Nice website, Quantum.</p>