- Level: International Baccalaureate
- Subject: Maths
- Word count: 3010
Mathematics Higher Level Internal Assessment Investigating the Sin Curve
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Introduction
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Investigating the Sine Curve
This report investigates the sine curve in the form o f, and how that relates to the graph of the sine curve. In particular, it would be investigated how the different variables ( effect the way that the graph is drawn and then seeing if the rule can be generalized to apply to any form of the equation.
The first thing to do would be to allow and to be 0, which would mean that the equation takes the form of: . It can be seen from Graph 1.1 that when is 1 what graph you get (the red graph) and when is allowed to be 2 what the graph looks like (the blue graph). From the graph below it can be seemn that increasing stretches the graph by the factor of change in . In simple words, the graph of would be twice the height of as is clearly seen from Graph 1.1.
When I change the value for , all the value of the sine curve get multiplied by that value of , which is 2 in this case. By multiplying all the values of the curve by the height of each point in the curve increases while there is no change in the position of the graph in the x-axis. For the original sine graph () it is common knowledge that the relative minima and maxima are -1 and 1 respectively, however when the is changed the original minima and maxima are also multiplied by and therefore the new minima and maxima would become and respectively. The rest of the graph also gets stretched by as this is the number that the whole curve is being multiplied by.
This time when we change the value of
Middle
In general it is observed that in the equation the period of the curve is given by . By changing the value of you can change the period of the graph. When you increase the the period of the graph decreases and when you decrease the the period of the graph increases. This means that the curve only stretches in the horizontal direction while there is no stretch in the vertical direction of the curve.
When you make and 0 you get the equation . In this equation, corresponds to the horizontal translation of the graph. When is a positive number the translation is a horizontal translation to the right, whereas when the value of is negative the graph translates to the left by units as illustrated by Graph 3.1.
When you change the value of you subtract the original value of by , and therefore the value of the new curve would be increased by to match the value. For example, when , where is 0, . However when there is a value for , cannot be 0 unless in which case must be equal to . This would translate the point and the rest of the graph by units to the right. This would only cause a translation meaning that there is no change in the amplitude or the period of the curve. From Graph 3.1, it can also be seen that that when is a negative number, the equation becomes and this causes the graph to translate units to the left.
From Graph 3.2 it can be seen that the value of can be any number. It can either be a fraction, a whole number or even an irrational number. The same concept follows for fractions and irrational numbers as well, where the value of
Conclusion
In conclusion, it was observed that there is a strong relationship between the variables of an equation and how the graph looks. In terms of the sine curve, the variables represent the amplitude, the period and the translations (horizontal and vertical) of the curve. This allowed us to come up with a graph without having to plot it, and when we plotted the graph our guess was shown to be correct indicating that the idea of the variables was correct. Near the end of the investigation, we used these variables and the sine curve to come up with a relationship between the sine and cosine curve which would then help us in future trigonometry calculations. Just like that, we can deduce different relationships between different sorts of sine, and cosine functions to help us in future calculations.
Greg McLean
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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