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Graphs illustrating variants of y = sin x.

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Introduction

Part 1  

image07.png

                                                                                                                                Graph 1

Graph 1 is showing y = sin x.

Now lets look at the graphs of y = 2sin x ; y = ⅓ sin x ; y = 5sin x

image08.pngimage13.png

 y =2sin x                                                    Graph 2     y = 1/3sin x                                                  Graph 3

image14.png

  y = 5sin x                                  Graph 4

If we compare graphs 2, 3 and 4 we can see that the number in front of sin (this number is called A) changes the vertical compression of the wave. When A<1 then the graph vertically compresses or amplitude becomes lower (graph 3) and when A>1 then graph expands vertically or amplitude becomes higher (graphs 2 and 4). If the number is 2, then the wave doubles vertically and when the number is ½ it compress by half. The comparison is of course made with the graph of sin x.  

Now let us see what happens when we make the equation negative by putting a minus sign in front of sin. By doing this we are taking A<0.

image15.png

                                                                                                                                 Graph 5

Graph 5 shows us that the wave flips around when A is negative. So we can conclude that when A<0 the wave will always be upside down

From investigating graphs of y = A

...read more.

Middle

B) will affect the wave.

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Graph 6

Here we have a graph of y = sin 5x compared with y = sin x. When B>1(graph 6) there is horizontal compression, but when B<1 there is horizontal expansion (graph 7). This is opposite to what we saw when we were exploring the graphs of y = Asin x.

image17.png

                                                                                                                               Graph 7

If we make B<0 then the wave flips around like did when we made A<0 (graph 5)

When B is for example 5, then the wave length will be the 5 times less than the wave length of sin x. When the B is ⅓ then the wave length expand 3 times the wave length of sin x.    

To conclude for the graphs of y = sin Bx, we say that when that is different from graphs of y = Asin x because y = sin Bx graphs expand or compress horizontally, while the graphs of y = Asin x compress and expand vertically.

Part 3

Now lets look at family of curves y = sin (x + C).

image18.png

                                                                                                                                Graph 8

...read more.

Conclusion

C = 3 which is greater than 0 and therefore it will move to the left by 3

Here is graph of y = ½ sin (3x + 3) taken step by step

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Prediction- y = -sin (½x - ½)

A= -1 which is less than 0, so therefore the wave will flip around

B= ½ which is less than 1, so the wave will horizontally expand by double

C= -½ which is less than 0, thus it will move to the right by ½  

Here is the graph of y = -sin (½x - ½)

image11.png

We can predict the shape and position of the graph for specific values because A,B, and C manipulates the wave and so we know that first we have to predict the graph of y = A sin x, then we use the graph we have and manipulate the graph with B. Now we have

y = A sin Bx and then we put in C and then we get the graph y = A sin B (x + C).  

 Part 5

image12.png

The function have similar shape and we can see that y = cos x is a transformation of

y = sin x. when we compare the two functions we see that the cosine function is shifted to the left by a half.

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...read more.

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Response to the question

This essay responds strongly to the task of investigating the sine function. For GCSE, this essay explores all the relevant transformations of the curve. Technical terms such as amplitude are used well, and show an understanding of the curve's nature. ...

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Response to the question

This essay responds strongly to the task of investigating the sine function. For GCSE, this essay explores all the relevant transformations of the curve. Technical terms such as amplitude are used well, and show an understanding of the curve's nature. If this essay wanted to go further, there could've been some exploration of the relationship between the sine and cosine curves. A simple discussion of how sin x = cos (pi/2 - x) would've been great here. If this essay was feeling extra ambitious, it could explore the summation of a sine and cosine curve, but this is a topic explored at A-Level.

Level of analysis

The analysis here is great. Each transformation is explained well, with a number of examples. The only way to fully show your understanding for a task such as this is to include graphs and sketches, and this essay does this brilliantly. It was great to see the exploration of -sin(x), as often candidates get fixed on looking at 2sin(x), 6sin(x) and so on. This shows the examiner that the candidate fully understands how the curve can be transformed. When looking at sin(Ax)
I would advise you mention the periodicity of the sine curve. Being able to discuss this in radians will prove you are an able candidate. The examples at the end show that this candidate is able to handle a number of transformations at the same time, which is often done poorly by GCSE students.

Quality of writing

The essay could've a few more technical terms such as reflection, periodicity, translation, enlargement as these are all key terms in mathematics when talking about transformation of functions. It would've been nice to see if the candidate could've handled the function notation, looking at f(x) = 2sin(x) and then drawing y = 2f(x - 7) + 3 for example. Spelling, punctuation and grammar are fine, and mathematical notation is good.


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