# EMT 668 - Algorithms & Computers

ASSIGNMENT 1, #5

by
### Kimberly N. Bennekin

### PROBLEM: Examine the graphs of y = a sin (bx + c) for different values
of a, b & c.

To begin the investigation, I will graph y = sin (x + 1) with **a**=1,
**b**=1 and **c**=1. This will be my base graph.

**y = sin (x + 1)**

In order to examine other sine graphs, I will vary **a**, **b **and
**c** individually and discuss the results. I will begin with **a**,
by substituting 2, 3, -1, -2, and -1/2.

### y = 2 sin(x+1) - red

y = 3 sin(x+1) - green

y = - sin(x+1) - blue

y= -2sin(x+1) - brown

y = -1/2sin(x+1) - purple

The graphs indicate that the numeric value of **a **will affect the
height of the sine wave. For **a**=2, the sine wave increases to 2 and
decreases to -2. For **a**=3, it increases to 3 and decreases to -3,
and so on. If **a** is negative, this creates a reflection about the
x-axis. By definition, **a is the amplitude of y = a sin (bx + c). **

I will do the same to **b**, by substituting 2, 3, 1/2, -1 and -2.

### y = sin (2x+1) - red

y = sin (3x+1) - green

y = sin(1/2x+1) - blue

y = sin(-x+1) - brown

y = sin(-2x+1) - purple

Changing **b** increases the intensity of the sine wave. By restricting
the domain of our graph to [-pi,pi], we notice that as **b** increases,
we increase the number of complete waves within this region. As **b **decreases
(as in y = sin(1/2x+1)), we decrease the number of complete waves within
this region. This happens because **b** directly affects the** period
**of the sine wave. The period of a trigonometric function is the distance
of one complete wave. The sign of **b** shifts the graph horizontally.
This brings us to **c** . I will substitute 2, 3, -1, and -1/2 for **c**.

### y = sin(x+2) - red

y = sin(x+3) - green

y = sin(x+(-1)) - blue

y = sin(x+(-1/2)) - brown

The value of **c** yields a** horizontal phase shift **of the graph.
If **c **is positive, the graph shifts **c** units to the left. If
**c** is negative, the graph shifts **c** units to the right. Recall,
y = sin(-x+1) gave us a horizontal shift (by substituting **b**= -1)
to the right. This happened since y = sin (-x+1) is equivalent to y = sin
- (x - 1), written in standard form. This allows us to see that** c**
= -1. This is why we have a shift to the right 1 unit.

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