This week in AP Calc, we talked about finding the slope of secant and tangent lines and recreating them ourselves online.
The activity at the beginning of the our enabled us to find the slope of four different lines which ended up forming the secant line. We were also given a variation of the point-slope formula, y=m(x-x ) + y , during the activity at the beginning of class which helped us figure out the equation of the actual secant line. The activity also helped us to better identify not only the secant line which crosses twice, but also the tangent line which only crosses once.
While making the first graph, we struggled with finding a way to make the secant line stay on the intersecting point. We didn't know what the point was supposed to be. After we plotted the initial point at (2,2), we needed a sliding point but just didn't know how to write it or where it should be. After talking to Mr. Cresswell, we figured out that the point must be random and represented by variables in order to be a sliding point. So, we first entered the expression given to us for the quadratic formula, f(x) = .5x^2. Next, we plotted the point (2,2), after that was when we had to identify a second sliding point. Now clear on what we had to do, we plotted the point (a, f(a)) which moved along the graph. Then we solved the rest of the question by graphing the secant line with the equation:
(f(a) - 2) * (x-2)
(a-2)
Now, with the second graph, we knew we had to follow a similar process. So, we graphed the same quadratic equation and plotted (a, f(a)). Now, we needed a second sliding point. We entered (s, f(s)) as our second sliding point. The last thing we needed was the equation for the secant line. We wrote down our answer and could not figure it out for a while, and then finally we talked to Mr. Cresswell and discovered that we had forgotten the y intercept in our secant line equation. Once we identified the problem, we wrote the equation:
(f(a) - f(s)) * (x-s) + f(s)
(a-s)
and finished our graph.
Our set-up worked well for the first two gifs because we used the same initial quadratic equation, so we could sort of emulate certain aspects of the first equation. The third equation asked us to challenge ourselves. It was actually the easier problem because we were able to leave all of our sliding points and secant lines the same, and enter any initial function for f(x), and it would work. The only difference that was present was the appearance of the graph.
Graph 1
Graph 2
Graph 3
The activity at the beginning of the our enabled us to find the slope of four different lines which ended up forming the secant line. We were also given a variation of the point-slope formula, y=m(x-x ) + y , during the activity at the beginning of class which helped us figure out the equation of the actual secant line. The activity also helped us to better identify not only the secant line which crosses twice, but also the tangent line which only crosses once.
While making the first graph, we struggled with finding a way to make the secant line stay on the intersecting point. We didn't know what the point was supposed to be. After we plotted the initial point at (2,2), we needed a sliding point but just didn't know how to write it or where it should be. After talking to Mr. Cresswell, we figured out that the point must be random and represented by variables in order to be a sliding point. So, we first entered the expression given to us for the quadratic formula, f(x) = .5x^2. Next, we plotted the point (2,2), after that was when we had to identify a second sliding point. Now clear on what we had to do, we plotted the point (a, f(a)) which moved along the graph. Then we solved the rest of the question by graphing the secant line with the equation:
(f(a) - 2) * (x-2)
(a-2)
Now, with the second graph, we knew we had to follow a similar process. So, we graphed the same quadratic equation and plotted (a, f(a)). Now, we needed a second sliding point. We entered (s, f(s)) as our second sliding point. The last thing we needed was the equation for the secant line. We wrote down our answer and could not figure it out for a while, and then finally we talked to Mr. Cresswell and discovered that we had forgotten the y intercept in our secant line equation. Once we identified the problem, we wrote the equation:
(f(a) - f(s)) * (x-s) + f(s)
(a-s)
and finished our graph.
Our set-up worked well for the first two gifs because we used the same initial quadratic equation, so we could sort of emulate certain aspects of the first equation. The third equation asked us to challenge ourselves. It was actually the easier problem because we were able to leave all of our sliding points and secant lines the same, and enter any initial function for f(x), and it would work. The only difference that was present was the appearance of the graph.
Graph 1
Graph 2
Graph 3