Other solutions could be generated by making some other vertical linethe axis of symmetry. It is also of interest to see both of the solutions on the same graph: So, a simpler case might be to let the slopes be 1 and-1, giving Again the tradition is to doso algebraically, but it might be instructive to look at some graphs ofh(x) - f(x) and h(x) - g(x), such as the following:Ĭan we immediately generate graphs of other f(x), g(x), and h(x) satisfyingthe conditions of the problem? Reviewing the graphs and the strategy, itsseems that the slopes of the lines can vary, the only condition being thatthey are m and -m. It remains to confirm f(x) and g(x) eachshare exactly one point in common with h(x). In fact, if the y-interceptswere equal, the y-axis would be the line of symmetry. One could try adjusting the y-intercepts. A zoom to the left hand sideshows a similar problem. How? Try making the slopes 3 and -3.The functions areĪ zoom to the right hand side of the graph with give It seems that the pair of tangent lineswill have to have this same symmetry. Looking back over the sequence of graphs(and perhaps generating some others) the graph of h(x) always has a lineof symmetry parallel to the y-axis. These graphs seem close, but clearly the line with negative slope isnot tangent to the graph of h(x). Still not too good but at least the graph of h(x) was "moved down." This is better? What can be observed? How can the graph of h(x) be "moveddown?" What if the graphs of f(x) and g(x) had smaller y-intercepts?Try One idea is to spread the two lines so that one has negative slope. This particular graph has one of the twolines "close" to being tangent to the product curve but the otherone is not close. Lets tryįor some novices, seeing the graph of the product h(x) = (3x + 2)(2x+1)and the graphs of the two straight lines from the factors on the same coordinateaxes provides a new experience. Senario OneLets open up the function grapher and explore with some specific f(x), g(x),and the resulting h(x). Experienced students, very bright students, and good problemsolvers could whittle this information and a lot more information out ofthis algebraic analysis. Thus,if there are points of tangency then they must occur at these common pointson the x-axis.
This means the graphof h(x) crosses the x-axis at the same two points as f(x) and g(x). This is useful becauseit keeps the students occupied, but what do they learn from it? Clearly,h(x) = (mx + b)(nx + c) is a polynomial of degree 2 and h(x) has two roots.The respective roots are when f(x) = 0 and g(x) = 0. A traditionalapproach would begin with algebraic manipulation. Is it possible to find two linear functions, f(x) = mx + b and g(x) = nx+ c, such that the function h(x) = f(x).g(x) is tangent to each. Further, the use of technologicaltools to examine visualizations of the functions makes for a different approachto the problem. In general it would not be included in the schoolcurriculum, but there is no reason it should not. Such tools make it possible to look at new topics in the mathematicscurriculum or to look at current topics in different ways. The senarios represent a compositeof several discussions of the problem with teachers, students, and colleagues.Note, the goal here is using this problem context, not only to solve theproblem posed, but to understand the concepts and procedures underlyingthe problem.įunction graphers are available for almost any computing platform or graphiccalculator. In fact, we came up with two different streamsof consciousness and so we have two senarios that are parallel in that theycover alternative approaches to the problem. Our analysis is presentedas a sort of stream of consciousness account of how one might explore theproblem with the tools at hand. This problem was posed by a group of teachers during a workshop in whichthe use of function graphers was being explored. In fact, this "paper" is a discussion of ourexamining a particular problem that having tools like function graphersavailable might make possible different approaches.įind two linear functions f(x) and g(x) such that the product This may have been an attempt to write a paper with a longer title thanthe paper itself. Each of which is Tangent to the Product Function by