The sine function is periodic and therefore, in order to gain a strong understanding of the curve, it is necessary to understand the important aspects of one period (a complete cycle) of the graph. To complete the following investigations, you should already have a solid grasp of the basic sine graph (y = sin x).

As a review of the basic sine function and a starting point for investigations into more complex sine functions, answer the following questions. Provide exact answers.

For y = sin x:* *** Click
here** to view the graph for help.

- What is the period and amplitude of the graph?
- What is the fundamental cycle of the graph?
- List the five guidepoints of the fundamental cycle. Identify the minimum, maximum, and x-intercepts.

This investigation uses the graphing calculator software, Graphing
Calculator 2.2.1. If you do not have this software, you can use
a demo version, or purchase the software through the internet
at **http://www.pacifict.com**.
This investigation can also be adapted to be used with other graphing
software, or with graphing calulators. Regardless of which graphing
technology is used, it is recommended that the graphing screen
be sized to show the fundamental cycle of the basic sine graph
(can found in link above), and then be resized, as necessary,
to best show the corresponding cycle of the "new" graphs.

Using your knowledge of the basic sine graph, you need to determine the shifts and translations that occur as the sine equation becomes more complex. By the end of this investigation, you should be able to describe the effects a, b, c, and d have on the function, so that general rules can be written. You should be able to use the knowledge gained from these investigations as a foundation for graphing the fundamental cycle of any sine graph. Remember, the key to understanding the shifts and translations is to compare and contrast the more complex graph with the basic sine graph.

1. What effect does a have on the graph? What mathematical term corresponds with a?

**Click here**
to open Graphing Calculator 2.2.1 as a helper application to view
graphs of y = sin x, y
= 2 sin x, and y = 3 sin x.

**Click here**
to view graphs of y = sin
x (purple), y = 2 sin x (red), and y = 3
sin x (blue).

Other values of a should be explored. Can you determine what effect -a has on the graph? Begin by exploring y = sin x and y = -sin x. Continue with opposite values of a until you feel like you have determined the effect the (-) has on the graph.

2. What effect does b have on the graph? What mathematical term corresponds with b?

**Click here**
to open Graphing Calculator 2.2.1 as a helper application to view
graphs of y = sin x, y
= sin 2x, and y = sin (1/2)x.

**Click here**
to view of y = sin x (purple),
y = sin 2x (red), and y = sin (1/2)x (blue).

Other values of b should be explored. Can you determine a mathematical relationship between b and the period of each graph?

3. What effect does c have on the graph?

**Click here**
to open Graphing Calculator 2.2.1 as a helper application to view
graphs of y = sin x, y
= sin (x+1), and y = sin (x-1).

**Click here**
to view graphs of y = sin
x (purple), y = sin (x+1) (red), and y = sin (x-1) (blue).

Other values of c should be explored.

You should have a strong idea of how c effects the graph, but it is important not to jump to an overall conclusion too quickly. b also plays a role in this type of shift. Apply your knowledge of b and c.

**Click here**
to open Graphing Calculatror 2.2.1 as a helper application to
view graphs of y = sin
2x, y = sin (2x+1), and y = sin (2x-1).

**Click here**
to view graphs of y = sin
2x (purple), y = sin (2x+1) (red), and y = sin (2x-1) (blue).

Other values of b and c should be explored. Can you determine the mathematical relationship between b and c and how this relationship effects the graph?

4. What effect does d have on the graph?

**Click here**
to open Graphing Calculator 2.2.1 as a helper application to view
graphs of y = sin x, y
= 1 + sin x, and y = -1 + sin x.

**Click here**
to view graphs of y = sin
x (purple), y = 1 + sin x (red), and y = -1 + sin x (blue).

Other values of d should be explored.

You have just completed a systematic investigation of sine graphs. Use your new knowledge to explain (in written words) all of the shifts and translations that take place in the following equations.

- y = sin (2x + 1)
- y = sin 2x - 1
- y = 4 + sin (x - 3)
- y = 3sin ((1/2)x - 4)
- y = -2sin 4x + 5

Organize the shifts and translations into a concise group of rules.

- amplitude:
- frequency:
- period:
- horizontal shifts:
- vertical shifts:
- interval for the fundamental cycle:

__Extension Investigations:__

1. Graph the fundamental cycle for 1-5 above.

2. Use these investigations as a model to analyze

If you have any comments concerning this investigation that
would be useful, especially for use at the high school level,
please send e-mail to **esiwdivad@yahoo.com.**

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