• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13

Mathematical Investigation

Extracts from this document...

Introduction

Type I: Mathematical Investigation

The equation of the sine function is defined as f(x) = a∙sin∙b(x+c) +d; f(x) is the function notation; a is the amplitude; sin is the function; b is the representation of the stretch factor; c represent any phase shift; d is the y intercept. (Any transformation of the sine function is compared to y=sin(x)).

Part I

Figure #1: The graph of y=sin(x)

image00.png

Figure #2: Comparison of different equations of sine functions: y=sin(x); y=2sin(x); y= (1/3) sin(x); y=5sin(x).

image01.png

Figure #3: Investigation of other sine graphs

image05.png

Based upon Figure #1 to Figure #3, as “a” varies only the amplitude of the graph changes according to |a|. Since only the value of “a” changed, there is no change in the period, no phase shift, no change in X and Y intercepts. As these figures illustrated, every single graph plotted has the same period (period is the horizontal distance between two points on the x-axis; period usually defined as the horizontal distance along x-axis of one complete cycle, 2pi radians) and the same X and Y intersects. There is no phase shift in any of the graphs because the position of the periods and x intercepts of each graph is the same.

Conjectures:

(a)

The transformation of the standard sine function y= a∙sin(x) by varying values of “a” only changes the standard sine function y= a∙sin(x)’s amplitude according to |a|.

...read more.

Middle

Part III

Investigate the family of curves y=sin(x+c)

Figure#5: Graphs of different functions of the family curve y=sin(x+c):y=sin(x); y=sin[x+ (pi/4)] and y=sin[x-(pi/4)].

image07.png

According to Figure #5, a change in the value of “c” shifts the entire function to the right or left of the origin; this is called a phase shift. When the red graph (y=sin(x) original) is compared to the blue graph [y=sin(x+ (pi/4)), the blue graph shifted 1/4 pi to the left while the length of each period and the amplitude remained the same with y=sin(x). When the red graph (original y=sin(x)) is compared to the green graph [y=sin(x-(pi/4)], the green graph is shifted 1/4 pi radians to the right while the length of each period and the amplitude remained the same with y=sin(x).

Conjectures

(a)

Based upon the observation above the transformation of the basic function f(x) = a∙sin (bx+c) +d with varying values of “c” is a horizontal translation along the x-axis; the direction of the translation is dictated by the zero of “c” in the angle (bx+c). For instance the zero of (bx+c) is –c, which means a horizontal translation along the x-axis to the left; vise versa if the zero of the bracket produces +c, the translation would be towards the right. The value of this horizontal translation is the vale of “c”.

...read more.

Conclusion

image02.png

According to Figure #11, the standard cosine graph y=cos(x) is a horizontally translated of the standard sine graph y=sin(x) to the left by pi/2 radians. To test this hypothesis the standard cosine graph is shifted pi/2 radians to the right in the equation y=cos(x-(pi/2)) and this graph is compared to the standard sine graph. If y=cos(x) is nothing but a horizontal translation of pi/2 radians to the left of y=sin(x), then sin(x) =cos(x-(pi/2)).

Figure #12: Graphs of the function y=cos(x-(pi/2)) and y=sin(x)

image03.png

According to Figure #12, the standard sine graph y=sin(x) = a horizontally translated to the right by pi/2 radians of the standard cosine graph y=cos(x). Based upon the Figure #11 and Figure #12, I conclude that cosine graph has the same amplitude and period/complete cycle as the standard sine graph; cosine is horizontally translated pi/2 radians to the left in reference to y=sin(x).

Based upon the properties observed two relations are obtained:

sin (x)=cos(x-(pi/2))

cos (x)=sin(x+(pi/2))

By combining the properties of the reflection on the x-axis and translation by the standard sine function y=sin(x), the standard cosine function y=cos(x) can also be obtained.

Figure #13: Graph of y=-sin(x-(pi/2)) and y=cos(x)

image04.png

According to Figure # 13, the reflection on the x-axis with horizontal translation to right (by pi/2) by the sine function is the same as y=cos(x). Therefore based upon the properties gathered from Figure #13, two relations are obtained:

cos (x)=-sin(x-(pi/2))

sin (x)=-cos(x+(pi/2))

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics 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 AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    x - x+h . 1 = x - (x + h) x + h x x+h x x(x+h) x(x+h) x + h - x h = x - (x+h) = -h = -1 = -1 = -1x-2. hx(x+h) hx(x+h) x(x+0)* x2 *h tends to 0.

  2. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    However, the process of OLS is a sampling method, the coefficient calculates will not be precise value. Thus, additional test is required to calculate the estimated value of the coefficient relative to its error. This test known as t-test which is very useful, the equation is : t=�/s.e.(�).

  1. Investigation into combined transoformations of 6 trigonometric functions

    be the combination of b which is stretching in the x-axis and c which is translating in the x-direction. I will be studying this combination with the trigonometric function secant which is 1 over cosine, it is shown by a purple dotted line in figure 1.0 A secant graph has

  2. Investigation of the Phi Function

    = ?(50) = 20 ?(5) x ?(10) = 4 x 4 = 16 Thus, ?(5 x 10) = ?(5) x ?(10) ii ?(7 x 6) = ?(42) = 12 ?(5) x ?(10) = 4 x 4 = 16 Therefore, ?(5 x 10)

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    Assume that the function representing time of sunrise in terms of day number has a period of 365 days. Explain how the value of parameter b could be determined algebraically. The value of parameter could be determined algebraically by determining the period, which is 365 days.

  2. Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    This means that if a person was calculating their consumption for period t they would use all the information available to them up until that time. However this can be broken down into areas; information available at time t-1 and "new" information that came available between t-1 and t.

  1. Sequences and series investigation

    Pos.in seq. 1 2 3 4 5 6 No.of squar. (c) 1 5 13 25 41 61 1st differ. (a+b) 0 4 8 12 16 20 2nd differ. (2a) 4 4 4 4 4 We can now use the equation an2 + bn + c 'n' indicating the position in the sequence.

  2. Estimate a consumption function for the UK economy explaining the statistical techniques you have ...

    The amount that the community spends on consumption depends (i) partly on the amount of their income, (ii) partly on other objective attendant circumstances, and (iii) partly on the objective needs and the psychological propensities and habits of the individuals comparing it...

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