#7. Let f(x)=a sin(bx+c) and g(x)=a cos(bx+c). Graph and explore for selected values of a,b, and c.

I first graphed both f(x) and g(x).

i.
h(x)=f(x)+g(x)

I noticed that the graph has max.'s and min.'s and the points of intersections of f(x) and g(x). This is because this is where there sums would be the greatest.

ii.
h(x)=f(x)g(x)
The graph that we are interested in has maximums between the other to graphs. We can account for the values of h(x) if we consider where they are coming from.

Though the graphs of both f(x) and g(x) where both consistent there appears at to be some inconsistencies when the to are multiplied together. Considering the multiplication of the two numbers resulting from the evaluation I am able to find a pattern in their multiplication,


iii
h(x)=f(x)/g(x)
The first graph looked like this

I really could not get what was going on with h(x). I kept increasing the bounds on the y.
Then I got this graph

Neat ! One can see the sin and cos. graphs that are real small. It is interesting how the division of two things that are so regular can look irregular at a first glance. But there is a rhythm to the graph
The following is a close up of f,g, and h . this is point where h is getting big. By examining the graphs you are able to tell why there is such a jump in the h graph. Both values of g(x) and f(x) are negative, which makes h(x) positive. The denominator is approaching 0 and the numerator is increasing between 1 to 2.

iv.
This graph did not have the extremes as the previous graph because we are
doing the function to the out put of the first function .

It might be helpful to look a graph of this equation by itself.

We notice that this graph does not go up and down but seems to skip a beat every other beat. This is because of the values being evaluated.

I believe that this problem was some of best work because of all of the investigations that I was able to do. I also like to look at cos and sin graph and had not considered the sum, product and quotion of the graphs.