This week we looked at Secant and Tangent lines. The activity at the beginning of the hour helped me understand the generalization represented in the first gif because it showed me the concept and how it worked mathematically, before we experimented with it and tried it on our own. Getting some background information on the concept helped me in understanding how to apply what we learned to make the different graphs on desmos. However, making the graphs on desmos wasn't all that easy.
First one took some time, but we managed to accomplish it before the end of the hour. The second one took us quite some time. We struggled with finding the secant line, even though it was a matter of us adding a y-intercept to the function. We also struggled with figuring out how to make the secant line connected to the quadratic function using endpoints and sliders. After we got more familiar with the actual program of desmos, we figured it out. We spent so much time trying to change the first graph to the second.
After finding out that the only changes that needed to be made were to replace the (x,y) coordinate (2,2) with 's' and f(s), and adding a y-intercept f(s), we were beating ourselves up for over thinking it. So our first graph was made up of: y=((f(a)-2)/(a-2))(x-2) ; f(x)=.5x^2 ; and (a,f(a)). Our second graph was made up of: y=((f(a)-f(s))/(a-s))(x-s)+f(s) ; f(x)=.5x^2 ; (a,f(a)) ; and (s,f(s)).
The gif we created was based somewhat off the first 2 gifs because we kept the secant line the same, and we also kept (a,f(a)) and (s,f(s)). The thing that we changed in the graph we created was instead of using a quadratic function, we used a cubic function. This changed the graph drastically. However, we kept the y-intercept part of the function. Analyzing the secant lines help us determine the tangent line of a function because it helps us determine the slope of the line.
Graph 1
Graph 2
Graph 3
First one took some time, but we managed to accomplish it before the end of the hour. The second one took us quite some time. We struggled with finding the secant line, even though it was a matter of us adding a y-intercept to the function. We also struggled with figuring out how to make the secant line connected to the quadratic function using endpoints and sliders. After we got more familiar with the actual program of desmos, we figured it out. We spent so much time trying to change the first graph to the second.
After finding out that the only changes that needed to be made were to replace the (x,y) coordinate (2,2) with 's' and f(s), and adding a y-intercept f(s), we were beating ourselves up for over thinking it. So our first graph was made up of: y=((f(a)-2)/(a-2))(x-2) ; f(x)=.5x^2 ; and (a,f(a)). Our second graph was made up of: y=((f(a)-f(s))/(a-s))(x-s)+f(s) ; f(x)=.5x^2 ; (a,f(a)) ; and (s,f(s)).
The gif we created was based somewhat off the first 2 gifs because we kept the secant line the same, and we also kept (a,f(a)) and (s,f(s)). The thing that we changed in the graph we created was instead of using a quadratic function, we used a cubic function. This changed the graph drastically. However, we kept the y-intercept part of the function. Analyzing the secant lines help us determine the tangent line of a function because it helps us determine the slope of the line.
Graph 1
Graph 2
Graph 3