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Investigating the Graphs of Sine Functions.

Extracts from this document...

Introduction

03.02.2004

Anna Markmann

Portfolio #2 - Type I

Investigating the Graphs of Sine Functions

The purpose of this assignment is to obtain general rules for transformations of sine functions from analysing patterns got from examples of these. To justify my conjectures of all of the following functions I used “Magic Graph”  - an electronic graphing program which allowed me to present them with a high level of precision. The trigonometric settings and the radian mode were kept constant throughout the whole investigation.

Part 1  

Graph ofy= sin x

To present this graph properly there are several possibilities: one can use a graphing calculator, a computer program, draw the graph from tabled values or from the unit circle. I chose the unit circle method because it is then more understandable how sine of x gets its shape and position, since sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse and therefore[1]:

image00.png

Characteristics of y= sin x

The sine curve is symmetric with the origin, it is an even function, and has infinite intercepts at multiples of image01.png, as well as infinite maximum and minimum points at –1 and 1.

Because the coefficient of sine is 1, the amplitude of y= sin x, the distance from the central value or the height of each peak above the baseline, is 1.

As the period of a function is the length of the time the system takes to go through one cycle of its motion the period of y= sin x is 2image01.png and the complete graph consists of the above graph repeated over and over: the domain of the sine curve is the set of all real numbers and the range is [-1,1].

Examples for y= Asin x (black graph)

and comparisons with y= sin x (grey graph)

Graph of y= 2sin x

image08.png

(a)

...read more.

Middle

Amplitude: the coefficient of sine is –1 and therefore the amplitude of

y= -1sin x is 1.

The period is 2image01.png, the complete graph consists of the above graph repeated over and over: the domain of the curve is the set of all real numbers and the range is [-1,1].

My conjectured graph of y= -sin x

image07.png

Justifying my conjectured graph of y= - sin x

image10.png

By comparing the true graph of y= -sin x and my conjecture of it, I can conclude that they are equal and therefore my conjecture for its transformations and its characteristics was right.

Conclusion for y= Asin x

As A varies in y= Asin x only the shape varies: A is the amplitude of the graph of sine. If A is negative the graph of y= -A sin x is obtained by changing the amplitude by its coefficient and reflecting it along the x axis.

Part 2

Graphs of type y= sin Bx

Graph of  y= sin 2x

Conjecture:

(a) transformation of the standard curve y= sin x:

as x is directly affected by the factor of 2 in y= sin 2x, I conjecture that the period of the standard curve, 2image01.png, is now multiplied by the factor of two: 4image01.png.

Thus y= sin 2x is a dilation along the x-axis by the factor of 2.

(b) characteristics of y= sin 2x:

the curve is symmetric with the origin, it is an even function, and has infinite intercepts at multiples of 2image03.png, as well as infinite maximum and minimum points at –1 and 1.

Amplitude: the coefficient of sine is 1 and therefore the amplitude of

y= sin 2x is 1.

The period is 4image01.png, the complete graph consists of the above graph repeated over and over: the domain of the curve is the set of all real numbers and the range is [-1,1].

My conjectured graph of y= sin 2x:

image07.png

...read more.

Conclusion

Graph of y= -sinimage23.png(x-1)

Conjecture: y= -sinimage23.png(x-1) is equal to y= -sin(image23.pngx-1/2) and is therefore obtained by dilation along the x-axis by the factor of 2, a translation of ½ units to the right along the x-axis and a reflection on the x-axis.

My conjectured graph of y=-sinimage23.png(x-1):

image24.png

Justifying my conjectured graph of y= -sinimage23.png(x-1):

image26.png

By comparing the true graph of y= -sinimage23.png(x-1) and my conjecture of it, I can conclude that they are equal and therefore my conjecture for its transformations and its characteristics was right.

Predicting the shape and position of y= Asin B(x+C)

The shape and position of y= Asin B(x+C) can be predicted since A is the amplitude of the graph, B the period of the graph and c determines the position of the graph.

A

 B

 C

Positive

dilation:

stretch along y-axis

 dilation:

shrink along the x-axis

 translation: shift along the x-axis C units to the left

 Negative

 dilation + reflection along x-axis

 dilation:

shift to the right along the x-axis

translation: shift along the x-axis C units to the right

Less than 1

Bigger than 0

 dilation:

shrink along the y-axis

 dilation:

stretch along the x-axis

translation: shift along the x-axis C units to the left

Part 5
Graph of y= cos x (black curve)

and y= sin x (grey curve)

image27.png

Both graphs have the same period (2image01.png) and the same amplitude (1) and therefore if y= A sin Bx and y= A cos Bx (Aimage28.png0, B>0) is image29.png the amplitude  and p= 2image01.png/ B the period.

 Just their position is different -  but if y= cos x is translated image01.png/2 units to the right along the x-axis y= cos x is equal to y= sin x and vice versa.

To justify all of the above a last example:

y= -sin(½x – ½) and y= -cos(½x –½)

image30.png


[1] http://documents.wolfram.com/teachersedition/Teacher/UnitCircleandSine.html

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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