# Mathematics Higher Level Internal Assessment Investigating the Sin Curve

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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 we would change it into a fraction, one fraction would be smaller than 1 whereas the other one would bigger than 1. It can be seen that when you have a fraction that is smaller than 1 for your value the height of the sine curve ill decrease to the height of . On the other hand, having a bigger fraction than 1 increases the height of the sine curve to the value as can be seen from Graph 1.2.

From these two graphs, it can be said that when is bigger than 1 the graph stretches outwards, whereas when is smaller than one the graph will stretch inwards. It is also seen that changing the of the sine curve changes the minima and maxima of the graph to . All the values of the sine curve are being multiplied by and therefore there is only a vertical shift and no horizontal shift.

We must not forget that could be a negative number as well and this has been explored with Graph 1.3 on the right hand side. In the sine curve if the becomes negative then all the values of the sine curve would be multiplied by the negative number. This would mean that all the positive values become negative and all the negative values become positive. To put this into simple terms the graph flips over with respect to the x-axis. Other than flipping over with respect to the x-axis, the graph stretches according to . In Graph 1.3 has been flipped over and stretched by a factor of to give the new graph of .

Overall, it can be seen that in the represents the ‘height’ of the graph. In mathematics this ‘height’ is known as the amplitude of the graph, meaning that in the equation above, represents the amplitude of the graph. When is negative the graph is not only stretched by but it is also flipped over the x-axis. This is happening because all the values of the graph are multiplied by which causes there to be a shift in the amplitude of the graph without affecting the horizontal position of the graph.