Assignment 1:

The Sine Function

by

Sarah Link

Examine graphs of **y = a*sin(bx+c) **for different values of **a, b, **and **c.**

The basic sine function, **y = sin(x)**, is where we'll start. The sine function is a cyclic trigonometric function that can be seen on the graph below.

So this occurs when **a=1; b=1; and c=0**. This graph has an amplitude, or height, of 1; a phase shift of 0; and a period of 2*pi. The phase shift is how much this original graph is moved left or right. The period is the length of the portion of the graph that gets repeated over and over again.

The first invegstigation we'll look into is changing the amplitude, which can be measured by taking half of distance between the minimum and maximum y values of the sine function. Now the question is, which value - **a, b, or c** - is the one for which the altitude is changed?

We know that to strech or compress a function vertically, we need to multiply by a constant; therefore we will first look to **a** to change the altitude of our graph.

Where **b** = 1 and **c** = 0

So as seen in the graph when the amplitude increasing, our graph becomes vertically longer. When we try to have a negative **a** value, we reflect our graph over the x-axis but we do not change the amplitude in any way.

The next value we want to examine the impact of is our **b** value. Instead of plotting multiple functions on a graph, we allowed **b** to be a variable value that ranges from -5 up to 5.

The animated graph below shows us that as when we changed **a**, the graph reflects over the x-axis when **b** becomes negative. When **b **approaches 0, the graph streches out horizontally and almost appears to pass through a straight line along the x-axis before reflected and then compressing horizontally.

Since we've investigated what happens when we change **a** and **b**, we now must turn to **c**. So far, we've kept **c** at a value of c. Let's let **c** = 1 and **c** = -1 to get a grasp on what might happen when we move those values even further away from zero. For these graphs we will keep **a** = 1 and **b** = 1.

We leave our original function in blue for comparison:

From the above graphs we can see that when we let **c** = 1 our graphs shifts left one unit; when **c** = -1 our graph shifts right one unit. We can make the conjecture that if we let **c** = 2 it will move 2 left and test this conjecture below:

As we thought, the graph has shifted by two units left. This move from left to right indicates our new phase.