• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematics Higher Level Internal Assessment Investigating the Sin Curve

Extracts from this document...

Introduction

 | Page

Investigating the Sine Curve

        This report investigates the sine curve in the form oimage12.pngimage12.png f, and how that relates to the graph of the sine curve. In particular, it would be investigated how the different variables (image38.pngimage38.png 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 image73.pngimage73.png and image35.pngimage35.png to be 0, which would mean that the equation image115.pngimage115.png takes the form of: image13.pngimage13.png. It can be seen from Graph 1.1 that when image15.pngimage15.png is 1 what graph you get (the red graph) and when image15.pngimage15.png is allowed to be 2 what the graph looks like (the blue graph). From the graph below it can be seemn that increasing image15.pngimage15.png stretches the graph by the factor of change in image15.pngimage15.png. In simple words, the graph of image39.pngimage39.png would be twice the height of image25.pngimage25.png as is clearly seen from Graph 1.1.

When I change the value for image15.pngimage15.png, all the value of the sine curve get multiplied by that value of image15.pngimage15.png, which is 2 in this case. By multiplying all the values of the curve by image15.pngimage15.png 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 (image25.pngimage25.png) it is common knowledge that the relative minima and maxima are -1 and 1 respectively, however when the image15.pngimage15.png is changed the original minima and maxima are also multiplied by image15.pngimage15.png and therefore the new minima and maxima would become image62.pngimage62.png and image15.pngimage15.png respectively. The rest of the graph also gets stretched by image15.pngimage15.png as this is the number that the whole curve is being multiplied by. image47.pngimage00.png

        This time when we change the value of image15.pngimage15.png

...read more.

Middle

In general it is observed that in the equation image31.pngimage31.png the period of the curve is given by image32.pngimage32.png. By changing the value of image14.pngimage14.png you can change the period of the graph.  When you increase the image14.pngimage14.png the period of the graph decreases and when you decrease the image14.pngimage14.png 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. image06.png

        When you make image34.pngimage34.png and image35.pngimage35.png 0 you get the equation image36.pngimage36.png. In this equation, image37.pngimage37.png corresponds to the horizontal translation of the graph. When image37.pngimage37.png is a positive number the translation is a horizontal translation to the right, whereas when the value of image37.pngimage37.png is negative the graph translates to the left by image37.pngimage37.png units as illustrated by Graph 3.1. image33.png

When you change the value of image37.pngimage37.png you subtract the original value of image16.pngimage16.png by image37.pngimage37.png, and therefore the image16.pngimage16.png value of the new curve would be increased by image37.pngimage37.png to match the image23.pngimage23.png value. For example, when image40.pngimage40.png, where image37.pngimage37.png is 0, image41.pngimage41.png. However when there is a value for image37.pngimage37.png,image42.pngimage42.png cannot be 0 unless image43.pngimage43.png in which case image16.pngimage16.png must be equal to image44.pngimage44.png. This would translate the point and the rest of the graph by image37.pngimage37.png 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 image37.pngimage37.png is a negative number, the equation becomes image45.pngimage45.png and this causes the graph to translate image37.pngimage37.png units to the left.  image07.png

        From Graph 3.2 it can be seen that the value of image37.pngimage37.png 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 image37.pngimage37.png

...read more.

Conclusion

image103.pngimage103.png looks just like the curve of image25.pngimage25.png and therefore could be a translation. Just like the sine curve, the amplitude (image15.pngimage15.png) of the cosine curve is 1, whereas, the period (image14.pngimage14.png) of the cosine curve is image104.pngimage104.png (just like sine curve), meaning that the image14.pngimage14.png value of the cosine curve would be 1. The only difference in the sine and cosine curve seems to be that the cosine curve has been translated to the left hand side by image26.pngimage26.png units, meaning that the value of image105.pngimage105.png and the value of image63.pngimage63.png. When we plug the numbers into the equation of the sine curve image31.pngimage31.png the equation that we get is the following: image106.pngimage106.png. Since this equation is for the cosine curve we can say that image107.pngimage107.png. However, there is another way to express this translation. It can be said that the graph of image25.pngimage25.png was flipped with respect to the image16.pngimage16.png axis and then translated horizontally to the right by image26.pngimage26.png units. When this is the case image15.pngimage15.png becomesimage108.pngimage108.png, image14.pngimage14.png and image35.pngimage35.png remain 0 and image37.pngimage37.png changes from image109.pngimage109.png to image26.pngimage26.png. The equation for this new translation would look like image110.pngimage110.png and is something else which could be used for calculations in trigonometry. image03.png

        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

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Lacsap's Fractions : Internal Assessment

    This proves that it is a quadratic equation. Therefore, the equation can be solved by using the Quadratic Regression 'QuadReg' function. The steps of this have been shown through a series of camera shots of the calculator. 1. Press the STAT button on the calculator, go to EDIT.

  2. Investigation of the Effect of Different types of Background Music on

    lowest number of words while listening to the music was a high dissonance level. The standard deviation of the words memorized for the control, low, medium and high dissonance was 1.93, 2.49, 2.77, and 2.81 respectively to three significant figures.

  1. Math Portfolio: trigonometry investigation (circle trig)

    Likewise, when the value of y is divided by the value of r, a negative number is divided by a positive number resulting to a negative number. The value of x equals a negative number in the quadrant 3 and the value of r equals a positive number as mentioned beforehand.

  2. Mathematics Internal Assessment: Finding area under a curve

    with area 0.76cm2, trap. (BCHF) with area 0.79cm2, trap. (CDIH) with area 0.85cm2 and trap. (DEJI) with area 0.95cm2. By increasing the number of trapeziums, I have reduced the area between the side of the trapezium and the curve f(x) = x2+3 [0, 1], thereby increasing the approximation of the area.

  1. Mathematics internal assessment type II- Fish production

    and (9, 527.8). Hence substituting the values: 9a3+9b2+9c+d=527.8 2a3+2b2+2c+d=450.5 6a3+6b2+6c+d=548.8 4a3+4b2+4c+d=356.9 Thus, solving the equations using the same Polysimul method on the GDC: Hence forming the equation for this interval: fx= 10.31x3-88.08x2+192.89x+334.53 Here is the equation of the curve against the graph: Graph for 1999 to 2006: As we can see in this

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    However, the linear function itself also is not without its weakness that it keeps linearly increasing and does not decrease its gradient; going up towards positive infinity; meaning that one-day should humans could be able to jump over 100 meters high should the trend be true.

  1. In this investigation, I will be modeling the revenue (income) that a firm can ...

    R= -Q2 + 6Q Finding the x-value 1. 2. = 1.8 1. 2. = 1 1. 2. = 3 Plugging the x-value to get the value for price 1. P = -1.25Q + 4.5 2. P = -1.25(1.8) + 4.5 3. P = -2.25 + 4.5 4. P = 2.25 1.

  2. In this Internal Assessment, functions that best model the population of China from 1950-1995 ...

    Parameter A represents the horizontal stretch of the graph. Parameter H represents the horizontal translation of the function. Parameter K represents the vertical translation of the function. The coordinate (-H, K) represents the turning point of the function. Regarding constraints, the cubic function for the population data would be limited to the first quadrant.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work