Section+1.5+Analyzing+Graphs

=Section 1.5 Analyzing Graphs= 8/25/11

Guess my function.
math

$ Mrs. Tyson displayed a graph on her TI-84, and asked us to guess the function. We guessed $f(x)=x$. Mrs. Tyson asked us to justify our guess, and we talked about things like: it's a line (how do you know -- constant slope). It has a y-intercept of 0. It looks like it is equidistant from the y-axis and the x-axis so the slope would be 1.

Mrs. Tyson asked what buttons could she push on the calculator to help us determine if we were right. We tried things like tracing -- the x and y coordinates seemed to match. We tried tracing to $x=1$, but the calculator returned an error as 1 wasn't in the current window. We looked at the table of the function starting at $x=0$ and going up with a step of 1 and we got a big surprise. The function increased for a while and then appeared to decrease. Therefore the function was not a line.

Thinking about the tracing and the table, helped us realize that Mrs. Tyson had zoomed in really far on the graph at $(0,0)$. When we looked at the function in ZoomTrig we saw that her function was $f(x)=\sin(x)$.

We used this experiment to talk about what it would mean to de{}fine the slope of a nonlinear function at a point. We felt that it would be reasonable to say the the slope of $\sin(x)$ at $x=0$ was 1.

Next we used Geometer's Sketchpad and we graphed $y=\sin(x)$. We placed two points on the function and constructed the line connecting them. We saw that when the points were far apart, the line did not resemble the function, but as we moved the points close together the line became a reasonable approximation for the function. Moving the points close together served a similar function to zooming in re{}peatedly on the graph.

Numerically we can move the points together by calculating the slope between $(x, \sin(x))$ and $(x+.001, \sin(x+.001))$. Using this numerical approximation for the slope of $\sin(x)$ at a point, we estimated the slope of $\sin(x)$ at. There was a lot of crankiness about getting all the 's in the right place on the calculator, but in the end we were able to see that the slope of $\sin(x)$ at the point $x$ is $\cos(x)$ because we recognized the decimals and Alex is going to put them in here for future use by the class. We could use the symmetry of the $\sin(x)$ graph to extend our slopes beyond the first quadrant. We also talked about the fact that the slope of a function at a turning poing is $0$. Pretty neat. $ math