# trigonometric functions

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Type 1 investigation

Transformation of Trigonometric Functions

Investigate the function:

y = a sin b(x-c) + d   in respect to the transformation of the base curve of y = sin x, depending on the values of a, b, c, and d. Be sure to consider all possible values for a, b, c, and d.

Describe the base curve

Then try values in b (sin2x, sin -3x, sin1/2x)

Does your hypothesis hold true for y = cos(x) and y = a cos b (x-c) +d? How about tan(x)?

Transformation of Trigonometric Functions

Introduction

The purpose of this study is to examine the transformation of trigonometric functions of y=A sin B (x-C) + D and determine the effect on the base curve y=sin(x). I am going to be systematically changing the values of A, B, C and D in the equation y= A sin B (x-C) +D. First I am going to examine different numbers for the value of A. I am going to use whole numbers, negative whole numbers, positive rational numbers and negative rational numbers for the value of A and see how this affects the Sine curve. Then I will examine different numbers for B, then C then D. After examining the Sine function I test to see if changing the values of A, B, C and D will have the same effect on the Cosine function.

Sine Curve:

Figure 1                                                                        Figure 2

I will use y-sin (x) as my base curve. This has a maximum value of 1, minimum value of -1 (which can be seen in figure 3 and 4) and an amplitude of 1, (which was found with the equation Amplitude= (max-min) / 2). The period of the base curve (which is the x-range, necessary to complete one oscillation) is 2 π. The sine graph repeats itself every 2 π (360 degrees), and this is seen in figure 2.

Figure 3                                                        Figure 4

Changing the value of the constant A:

I am now going to change the values of A in the equation y= A sin to see what affect it has on the base curve.

y= 3sin(x)

Figure 5

From figure 5 we can see that placing a 3 for the value of A has transformed the base curve. The amplitude of the graph is now 3, the minimum value is -3 and the maximum value is now 3. And the period remained 2 π.

y= -3sin(x)

Figure 6

Figure 6 show that a negative whole number also changes the amplitude and maximum value to 3, and the minimum value to -3. The period of this graph still remains 2 π. But using the trace button on my graphing calculator, we can see that graph is inverted. I predict that if I replace A with a value smaller than 1, than the amplitude will decrease from 1. And if A has a negative value then the graph will invert.

y= 0.25sin(x)

Figure 7

In figure 7 it can ...