tag:blogger.com,1999:blog-6651514617266100245.post861937075684335976..comments2023-05-28T09:46:04.615-04:00Comments on I Hope This Old Train Breaks Down...: Beyond the Algebra of Composition FunctionsUnknownnoreply@blogger.comBlogger7125tag:blogger.com,1999:blog-6651514617266100245.post-38570253899871599332012-03-06T19:22:43.503-05:002012-03-06T19:22:43.503-05:00Just found this blog! I'm a student teacher a...Just found this blog! I'm a student teacher and this will help a lot - thanks for your great ideas!Lauriehttps://www.blogger.com/profile/04444815132849713694noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-45699415663249082722011-04-11T13:50:44.479-04:002011-04-11T13:50:44.479-04:00No problem. Thanks for leading me back to this pos...No problem. Thanks for leading me back to this post. I didn't realize that what before were just some dollar sign typing shortcuts had become error statements upon my recent installation of LaTeX into blogger.<br /><br />Funniness. I fixed it now so the entry would make sense again.untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-82362994345773289212011-04-11T09:17:53.831-04:002011-04-11T09:17:53.831-04:00Thank you for this post! I will try the activity w...Thank you for this post! I will try the activity with the "secret basic operation" in my Summer Enrichment class.:) It's a really good motivational activity.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-76398354721012447282011-03-12T08:23:11.085-05:002011-03-12T08:23:11.085-05:00"Fairness" could work. I'm thinking ..."Fairness" could work. I'm thinking more along the lines of "laziness"...<br /><br />In my original post about function transformations (see http://untilnextstop.blogspot.com/2011/02/function-transformations-nitty-gritties.html ), another reader ("glsr") had described the transformations as something that happens to the axes instead of to x or y values. For example,<br /><br />y = 1/3 sin(2x-1) - 5, that means that the axes are working hard to double the x's and subtract 1. So, to get to the same location in the graph as originally, the x's get to hang out, go at half speed, and just add 1 to start. I think that's a nice combo of both your explanation and his. He also explains using the same way for y, except that that looks like 3(y + 5) = sin(2x - 1) if you move the y numbers to the "correct side." So yes, each y value also gets to chill out. They get to go at 1/3 speed, and start 5 units less than usual (ie. 5 below 0), since the y axis is aleady doing so much work for them.<br /><br />Thoughts?untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-71240487907479445362011-03-11T19:59:17.896-05:002011-03-11T19:59:17.896-05:00Hmmm...maybe (playing off inverse functions a bit)...Hmmm...maybe (playing off inverse functions a bit) talk about it like a race--we want everyone to be on a fair level at the start of the race (before we input into the function, whether it is a sine, quadratic, square root, etc), but after the start of the race (when the main function is performed) you can do whatever you want. <br /><br />For example: y = 4sin(x + 1) + 3. Before we go into the race, make everything fair--this guy is already 1 ahead of everyone, so we need to back him up one to be fair. But after the race starts, not only does he get to move up 3, but he also gets to multiply all his values by 4. <br /><br />y= 1/3 sin(2x-1) - 5 This guy looks like he's behind by one, but since he moves twice as fast, he's only behind by 1/2. However, since he does travel twice as fast, we better cut all his x values in half to be fair. After the race starts, he only goes up 1/3 high as everyone else, plus sadly he moves down 5. <br /><br />Plus here is the website I was talking about--scroll down for the piston http://www.intmath.com/trigonometric-graphs/2-graphs-sine-cosine-period.php<br /><br />Does this make sense? I know it's not a really "mathy" explaination. I'm kind of thinking out loud here, but I'm about to do trig graphs soon so it's good to think about. :)Meghttps://www.blogger.com/profile/08395474750276931370noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-71856833793359271142011-03-10T21:54:32.300-05:002011-03-10T21:54:32.300-05:00Thanks, Meg!
I like your twice the speed idea, b...Thanks, Meg! <br /><br />I like your twice the speed idea, but how does that work with the horizontal shifts?<br /><br />Mimiuntilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-89159461334458940602011-03-10T19:45:43.919-05:002011-03-10T19:45:43.919-05:00Love your telephone idea!! Definitely adding that...Love your telephone idea!! Definitely adding that to my list of cool things!<br /><br />Also, in reference to a blog post a while back, I finally figured out a good way of explaining why, if you have 2x, you 1/2 all the x values for graphing. Just looking at a linear function, if you go at twice the speed (2x), it will take you 1/2 the time to get there. There is a good piston/sine function java doohickey around the interwebs I will try to find again that gives a really good visual representation of this.Meghttps://www.blogger.com/profile/08395474750276931370noreply@blogger.com