The graphing calculator is one of the most powerful tools we use in mathematics today. HOWEVER, it is not more powerful than the human brain despite what the average student today might think. While it is a great too, it cannot replace our basic understanding of numbers and graphs and it is not a substitute for common sense! To demonstrate this I've put together a collection of "tricks" on the calculator that will test whether you believe everything the calculator tells you or you have some sense of your own!!
Consider the task of graphing a simple trig function. We are going to graph this in DEGREE mode but we could just as easily run this exercise in RADIAN mode. The only task I am asking is that you graph the function y=sin(45x) and make an analysis of the graph of that function. Now, lets say you went to your calculator for help...you graphed the function as usual and set up a standard ZOOM TRIG as a way to view the graph. What conclusions can you make about the graph?What do you think about this graph? On the surface it looks very nominal...sine in nature, bouncy, curvy...all the good things. However, consider how many times this graph should repeat itself given the "b-value" of 45. Also consider the slope that a standard sign graph should have at the origin.
The problem becomes compounded when we start to change the window settings to try and fix this problem. Say for example we change the x-values of the window to [-350,370] as shown below. What is supposed to be a regular sine graph looks even worse!!
Notice anything strange about the x-intercept? Similar examples are shown below with graphs of some other trig functions that aren't all that they appear or all that they should be!
sin(46x) on [-360,360]sin(47x) on [-359,361]