Write Up 1

by Jamila K. Eagles

An eploration of the graphs of sine and cosine functions.
Assignment #1 problem # 7

Let f(x) = a sin(bx + c) and g(x) = a cos (bx + c)

For selected values of a, b, and c, graph and explore:

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

ii. h(x) = f(x) * g(x)

iii. h(x) = f(x) / g(x)

iv. h(x) = f(g(x))
Let a= 1, b=2, and c=0

In this case, f(x) = sin 2x and g(x) = cos 2x. The sine function is graphed in red and the cosine function is shown in green in the graph below. Both are functions that have an amplitude of 1 and -1.

Now let's see what happens when we add the two functions. When adding f(x) + g(x) we get the equation y = sin 2x + cos 2x. It is represented by the blue curve in the graph shown below. Adding the two functions gives us a curve with a larger amplitude. Notice that the highest and lowest points of the curve occur where the sine and cosine function intersect.

Next we multiply the two functions. We now have a new equation f(x) * g(x). The new curve is represented by the blue curve in the graph below. We see that the amplitude is smaller. We also notice that the highest points of the curve occur where the sine and cosine functions intersect. Each time the sine and cosine function intersect both above and below the x axis, our new function reaches its highest point above the x axis. We also notice the new function repeats its pattern more times than the individual functions. This shows us that the the period also decreases.

We can now divide our functions. We have . The new function is represented by the curve shown below in blue. This new graph is known as the tangent function . What we have graphed is tan 2x. The tangent curve is formed with vertical asymptotes on each side. The curve is tangent to the x axis for each seperate cycle. Meaning that it only crosses the x axis at one point per cycle. From the graph below we also see that the curve is tangent to the y axis, only crossing at y = 0.

The function f(g(x)) gives us another type of curve shown below in blue. The graph is a representation of the equation, sin 2(cos2x). We see that the new function appears to take on a more square shape. It has the same period as the cosine function. It appears to take the same shape as the cosine function, but at its peak it has a dip, forming two peaks.

Comparing the functions of sine and cosine , adding the functions, multiplying , dividing , and having one the function of the other is a discovery that can help students see what happens to the graphs when the functions are manipulated.




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