# 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.

Moving on, we now set  and  at 0 giving the equation  the form of . It is observed that changing the  has the same effect as changing the  in the equation, except rather than the curve stretching vertically it stretches horizontally. This is known as a period in mathematics. It is the interval between likewise values, or in simpler terms the interval before the graph ‘repeats’ itself. The graph below shows the curve stretching inwards or outwards (depending on whether the  value is greater than or smaller than 1) without changing the vertical position of the curve.

The reason ...