<p>The figure above shows the graph of the quadratic function with equation y=ax^2 + bx + c, where a, b, and c are constants. WHich of the following is true about a?</p>
<p>I don't really have the photo of the function but the only point that there is on the function given to you is -2, -1. </p>
<p>A) A < -1
B)-1 < a < 0
C)0<a<1 d)a="1" e)a="">1</a<1></p>
<p>You cannot solve it without the photo. In fact, the photo can help you to know if a>0, and something about c. Hence you can solve it.</p>
<p>We need a picture to solve it</p>
<p>Is -2, -1 the vertex? Is the function facing down or up?</p>
<p>It says the figure above dude. where is the pic ?</p>
<p>I have such a figure, but don’t know how to post it here.</p>
<p>Actually, it’s easy to solve this problem.</p>
<p>Noting that (-2, -1) is the vertex, we assume the quadratic equation ax^2 + bx + c = 0 has two roots: -3/2 and -5/2.
So f(x) = a(x + 3/2)(x + 5/2).
Recalling f(-2) = -1, we get a(-2 + 3/2)(-2 + 5/2) = -1.
Then a = 4.
Answer is (E).</p>
<p>Second approach:</p>
<p>Write the function in vertex form:
f(x) = a(x+2)^2 - 1.
Thus, f(x) = 0 has two roots:
x<em>1 = -2 + sqrt(1/a)<br>
x</em>2 = -2 - sqrt(1/a).
From the figure, we have:
x_1 = -2 + sqrt(1/a) < -1.
So,
sqrt(1/a) < 1.
Therefore,
a>1.
Answer is (E).</p>
<p>If I remember this problem correctly…</p>
<p>For a parabola that has a=1, when you go from the vertex to the left or right by one unit, the graph goes up unit. But you can see from the graph of this particular parabola that it goes up by more than one unit. So a>1.</p>
<p>(But again, I may be thinking of a different problem altogether…)</p>
<p>Why you started from a =1, rather than for example from a = -1?</p>
<p>The parabola in the picture opens upward so you know a is positive. The only issue to resolve is whether it is more, less or equal to 1. And you can tell which is the case by looking at how steeply it rises. From the graph, you can see that when you look one unit left or right of the vertex, it rises lots more than 1…</p>
<p>The SAT actually asks very little in terms of specific knowledge about parabolas:</p>
<p>Know their basic shape and symmetry.</p>
<p>Know that the ‘a’ value controls steepness, its sign controls whether it opens up or down. </p>
<p>The ‘c’ value controls y-intercept.</p>
<p>The ‘b’ value, together with the ‘a’ value determines vertex location. But it is a debatable point as to whether -b/2a is worth knowing…some say yes though I haven’t seen the problems that require it…no harm though knowing it.</p>
<p>@pckeller</p>
<p>I’ve never seen a problem that absolutely requires the formula x=-b/(2a), but I have seen at least one question where it gives the quickest solution.</p>
<p>In my opinion anyone going for an 800 (or high 700s) should definately know this formula. Yes, there is only a very small chance that it will be useful on any given SAT, but this is true about a lot of the things I teach to my higher scoring students.</p>
<p>Cool. I suggest we do add the Steve’s Formula to our CC arsenal. I got such label for the average speed formula (which I only borrowed from a good source.) It’d be nice to have a DrS or a PCK set of formulas. :)</p>
<p>“Xiggi’s formula” has already transcended CC. I no longer use the term “harmonic mean” or “average speed.” I always say Xiggi’s formula when I’m teaching, tutoring, posting, writing, etc. Other SAT tutors that use my materials are using this term as well since that’s what I have printed there. I expect that in about 50 years “Xiggi’s formula” will be the international standard. :)</p>
<p>Thanks for the comical interlude. </p>
<p>By the way, to be clear, I was not sarcastic and I take this opportunity to thank the professional tutors and teachers for their efforts on this site. I am sure that many have found the repeated contributions most helpful.</p>
<p>Despite the tone and the emoticon at the end of my post, I was not being sarcastic either. I only added the Harmonic Mean formula to my list of strategies for advanced students after seeing you emphasize it on CC. I now realize that it’s a powerful formula in the rare instance when a specific type of problem shows up. So there are two reasons I use the name “Xiggi’s formulas” - first because I think it’s a catchy name, and more importantly because although you didn’t create the formula, you are directly responsible for my current use of it with strong SAT students.</p>
<p>After I “discovered” Xiggi’s formula here I also made it a point to look at lots of Level 5 SAT problems and create a more extensive list of advanced strategies, formulas, etc. that are useful infrequently, but that I feel are helpful for students going for an 800. </p>
<p>And by the way, x=-b/(2a) is on that list.</p>
<p>I’ll probably start a thread to discuss these soon.</p>
<p>OK: In the spirit of compromise, I will include -b/(2a) into the canon (I bet you didn’t know that I am in charge of the canon. It’s a side gig I picked up…) But in return, we have to put back 7,24,25 and 8,15,17 along with 3,4,5 6,8,10 and 5,12,13. </p>
<p>Also, another game would be to try to identify things that are JUST outside of the canon – things that have never been tested but are so close to SAT-like that they are worth mentioning to top scorers just in case. For example, the graph of |f(x)| is occasionally tested. But I’ve never seen the graph of f(|x|) tested. And yet…</p>
<p>By the way, where did you get the SAT Jan 2013 math questions?</p>
<p>I suppose we can put those Pythagorean triples in - I guess that means I’ll have to memorize them!</p>
<p>I really like the idea of identifying things just outside the canon. And actually we don’t know if these things have been tested or not. It’s quite conceivable that the graph of f(|x|) has been tested on an unreleased SAT. It’s probably worth starting a thread on this.</p>
<p>By the way here is a quote taken from a pm to me from a student on this forum:</p>
<p>“I believe you helped me with the (-b/2a) formula of the parabola… I just wanna say thanks… I wouldn’t have gotten my 800 without it.”</p>
<p>He’s referring to the January 2013 SAT here. I haven’t seen this test, so I’m not sure which question.</p>
<p>@DrSteve</p>
<p>I think it’s like the whole nCr debate: there are problems where you COULD use -b/(2a) but no problems where it is the only way (or even the shortest way). Just this evening, I reviewed the January test with a student who had the QAS. The only parabola question is the one that started this thread. So I don’t know which one your student could be referring to. Until I hear otherwise, I still call it “Just outside canon.”</p>
<p>BTW, looking over that QAS, now I know where a whole bunch of recent threads got their start!</p>