You may want to read the comments for this post. Answers and possible solutions are discussed. There is also considerable discussion about teaching techniques for f(x-h).
The parabola problem from 2-15-07 generated some interesting discussion. I haven't had a chance to see it implemented with our Algebra 2 classes yet but I'll let you know if and when...
Today's problem is along the same lines. I'm trying to provide some problems that are exclusively high school math content for this time of year. There are dozens of outstanding problem-solving sites for MathCounts and similar middle-school competitions but there appears to be a dearth of secondary math problem-solving sites (or I haven't found them yet!). Again, how might one use the problem below? As a bonus or an extended in-class activity or a performance assessment or ??? How many would regard this question as suitable only for honors or accelerated students? My take is that if students are exposed to higher levels of thinking and know they are expected to learn how to do these and held accountable on an assessment, they will adjust. Not all will experience equal success but that's ok too! Many should be able to do part (a) or are my expectations way too high?
(a) Consider the quadratic function f(x) = 4(x+4)2.
The graph intersects the line y = k, k>0, in 2 distinct points B and C.
The rectangle whose base is on the x-axis and 2 of whose vertices are B and C
has area 64. Determine the value of k. Show method clearly.
(b) Now let's generalize the result of (a).
Consider the quadratic function f(x) = a(x-h)2, a>0.
The line y = k, k>0, intersects the graph of f in two distinct points B and C. The rectangle whose base is on the x-axis and two of whose vertices are B and C has area R.
(i) Explain graphically (not algebraically) why the area, R, of this rectangle is independent of h.
(ii) Express k in terms of a and R. Check that your formula for k gives the value you obtained from part (a).
Friday, February 16, 2007
Another Quadratic Function Problem 2-16-07 through 2-20-07
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13 comments:
For high school level problems, you might want to check out:
AMC Practice Problems and
AMC Topical Quizzes
Also
The Art of Problem Solving forums have tons of high school level problems for all abilities, and they have archives of many high school level (and beyond) contests.
I always had trouble getting my students to understand that replacing x with x−k in an equation moved the locus k units to the right. They think that subtraction moves things to the left.
dear mathmom--
i am very aware of the amc and AoPS sites -- i use them to prepare our students and I have been very impressed with their Math Jams. Some of their questions tie nicely to content that we teach but much of it is perceived by some teachers as 'contest problems'. I'm trying to develop materials that can be used by the classroom teacher for regular college-prep classes to develop topics further, provide richer explorations, serve as models for assessments. etc. There's a fine line here but the format of the last 2 parabola problems I've written are significanjtly different from contest types. They have several parts and are of increasing compleixty.
In other words, if a math teacher googles 'quadratic functions activity', they are looking for ideas to use with all students in their class. Therefore, I want the questions to reflect typical concepts and skills taught in this unit. I don't see these kinds of problems in most textbooks. Some might appear in ancillary materials like Enrichment Materials, which means they are often overlooked! The abstract parts of these problems seem a bit ambitious for the typical student but we need to expect more from our youngsters, not less. Of course you may be coming across these on the sites you've mentioned in which case I stand corrected!
eric--
you motivated me to start a new thread:
Most successful ways teachers have found to teach transformations like f(x-h)
Of course, the truth is most methods have failed and students just remember to do the opposite of whatever the sign says!
Ok, I'll start with a couple of my favorites:
1. Focus on equations first, which result naturally from finding the x-intercepts. Thus, y= x^2 - 1 leads to the equation x^2 = 1 and solutions (x-intercepts) x = 1 and -1.
Now consider the function y = (x-3)^2 - 1. The x-intercepts lead to the eqn (x-3)^2 = 1. When solving this by taking square roots, we obtain:
x = 1 + 3 and -1 + 3. Note that the inverse operation becomes very evident. I ask the students to describe using transformational language what happened to the original x-intercepts: TRANSLATION, 3 TO THE RIGHT!
2. Focus on domain. Consider f(x) = sqrt(x) which has domain x>=0. Now consider g(x) = sqrt(x-3). We obtain the domain by working with the 'inside' function 'x-3'. Setting x-3>=0 leads to x>=3. Again students experience a translation of 3 units to the right. This is even more dramatic when using the TABLE feature on graphing calculators. Student can see the domain NUMERICALLY shifted to the right. The graph shows it visually and of course that is powerful, particularly when the original function is graphed FOLLOWED by the translated graph in bold mode. Ok, enough, now I invite others' favorites. Hey, we all can learn here since this is such a challenging concept for many.
Any bets that this thread will die a bornin'....
Dave,
I don't think there is as much distinction as you think between "classroom problems" and "contest problems". It is true that the problems you have created with multiple steps are not usually found at "contest" sites, and they are great, no argument here! But I'd think a teacher could find plenty of appropriate problems on "contest" sites. They need to adjust what they find just about anywhere to fit the needs of their particular class, and they could do this with "contest" problems too.
All this is in no way meant to minimize the value of what you're doing here. Just that there are other places high school teachers could start looking for problems. On the AMC site, they have a database arranged by NCTM standards, even! They are not just for contest prep.
One of the reasons the principal/math teacher at the school where I volunteer lets me take regular classroom time to work on "contest problems" with all the kids (K-8, and not just those who are advanced in math, and not just those interested in math contests) is because she recognizes that the depth of mathematical thinking and problems solving skill that comes from working on problems like these is worthwhile and will help both their mathematical and overall intellectual development long-term.
Dave, I taught a teacher once who claimed to have success teaching (x+h) shifted the plane h units to the right. Nice guy, but I hardly think I believe him.
He didn't believe me either. We went for a picnic one day, and I was sitting quietly, munching an ear of corn. He came over and I, without looking up, announced that this ear had 18 rows. Nothing. And then I announced that all ears of corn had an even number of rows, at which point he said "2718, you are full of s____"
Since I was reading "The Staffs of Life" and had it with me, I opened it to the even-rows page...
Anyhow, I start emphasizing subtraction to measure distance long long before we have any y=a|x-h|+k sort of stuff. It helps, but it's no golden bullet.
Oh, yeah, the problem. The vertex is at (-4,0) So find a point such that 2(x--4)(y) = 64
2[4(x+4)^2)](x+4) = 64
(x+4)^3 = 8
x = -2
B = (-2,16)
y = 16, k = 16.
for the second problem, h just shifts us left and right, without altering the distance between B and C (nice hint, thanks)
WLOG, consider f(x) = ax^2.
The area of the rectangle will be the height of the point on the parabola, k, times the width, 2ak^2
R = 2ak^3.
k = (R/2a)^(1/3)
In one of our Algebra II classes (I am not teaching that course), a teacher taught paper-folding conics this week... I am curious to see how the kids do with them. She's braver than me: when I last taught the course I shied away from any in-depth work with foci and directrices.
Dave, I don’t know if you’ve seen this, but it’s something to consider:
Math for the Masses at WordPress.com’s official blog. I also wrote a post about it.
As far as I can tell, it only supports the inline mode (at least right now).
Display-style LaTeX has now been enabled, as has LaTex in comments at WordPress. Please see my weblog post, including the comments for a demonstration.
If Blogger does not offer something similar soon, you may wish to give some serious thought to switching services—just thought you’d be interested.
jonathan--
very nice solution for (a) using the symmetry around the vertex which allows us to represent the area in terms of (x+4) rather than k! actually, you don't need to solve for x, since k = 4(x+4)^2 =
4(2)^2 = 16;
in part (b) I believe you accidentally used k in place of x, so actually you solved for the x-value rather than k. Just plug that back into the function to obtain the value of k: (aR^2/4)^(1/3).
Also, regarding that math teacher---
In the old days, moving the plane h units was actually taught in a course called analytic geometry i took in college. However, as I recall it, it wasn't stated that the plane was moved. Instead of translating the graph, say 3 units to the right, we would keep the curve where it was and move the y-axis 3 units to the left! This was explained by the phrase 'transformation of coordinates' (later on we used vectors and matrices to describe this). If we make a new y-axis at x = -3, then every point on the graph has the same y-coord but NEW x-coordinate as follows:
Using the equations of the y-axis,
x_old = -3 becomes x_new = 0, therefore
x_new = x_old + 3. For example, (2,4) is on the graph of y = x^2. Here x_old = 2. Relative to our new y-axis (transforming coord),
x_new = 2+3 = 5, so the SAME point now has coordinates (5,4) under our NEW coord system.
In other words, when analyzing the new function, y = (x-3)^2, we would express the transformation as follows:
Think of "x-3" as "x" so that
x_old = x_new - 3 or x_new = x_old + 3. The "+3" is why the graph can then be thought of as sliding 3 to the right.
I believe this actually made sense to me at the time after I thought about it deeply! Besides, have you ever messed up a graph and had to erase either the x- or y-axis and reposition it to compensate rather than re-draw the curve!
Now how many students will appreciate this!!
mathmom--
i agree with you that contest problems can be modified for many students and could be very useful to 'raise the bar'. thanks for helping me rethink that!
finally, darmok--
thank you so much for the LaTeX info. I went to your site and love it! I definitely think I can pick up LaTeX even if the pronunciation is unusual! I've been feeling for a while that i made a mistake using Blogger and that WordPress meets my needs far more. But I am not sure how to simply move my files over without losing them or my readers. I'm guessing that wordpress will redirect viewers to my new site but I'm not certain of that. Further, Blogger displays images their own way so some of my jpg files may not view correctly in wordpress.
Dave,
open a wordpress account, move evertything over, leave the blogger account up.
You can replace the blogger with pointers to your new account if/when you are satisfied with what you have at wordpress.
I'd say that especially in your case, since you have so many equations and expressions, that wordpress just became a much better option.
And yup, I messed up k and x. And I did part I without the hint. Didn't even think about h dropping out.
See what I did? I engaged with the first part without reading the second. We know that students should read the whole problem first... but we know how often they (and I, and maybe more of us) do not.
THANKS JONATHAN!!
I think I'll play with LaTeX and see what it looks like for a while. Wordpress supplies these pointers automatically? Some readers may not be directly visiting my site - rather they are just subscribing to feeds. Will that also be switched over?
Dave
You’re welcome, Dave; the addition of LaTeX was unexpected (for me) but exciting nonetheless. You have to play around with it for a bit but I picked up the basics pretty quickly, I think. What WordPress uses is similar to the inline notation you'll see: for instance, you could type $latex \phi = \frac{1+\sqrt{5}}{2} $. Backslashes precede commands/functions, and curly braces are the grouping symbol.
I don't have any direct experience with importing from other services, but apparently it’s relatively simple now, according to this. One caveat, though, is that images apparently must be copied manually.
What you’ll have to do is once (if) you switch to WordPress, post an entry on Blogger alerting your readers of the switch, and provide the URL for your new feed. They'll have to update their feed readers.
If I can be of more help, just ask!
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