Trigonometric Functions: Transformation.

Aims: We will investigate the properties of some trigonometric functions:
f(x) = a sin b(x - d) + c, f(x) = a cos b(x - d) + c, & f(x) = a tan b(x - d) + c.
We will also pay special attention to the concept of amplitude and period.

  1. The diagram at the side shows y= sin(x) and y=cos(x). The "angle" x is measured in radian.
  2. The period is the distance on the horizontal axis (in our case x-axis) between "identical" (refer back to the unit circle to understand the meaning of "identical") places on the graph. For simplicity, we usually think of period as the distance between successive peaks or troughs. The period for y=sin(x) and y=cos(x) are both 2π .
  3. The amplitude of a trigonometric function is the distance between the principal axis (in our case the x-axis) and of the maximum or the minimum point.


    Questions
    1. What is the amplitude of y=sin(x)? __________________

    2. What is the amplitude of y=cos(x)? __________________

    3. What is the period and amplitude of y=tan(x)? __________________

    4. What is the maximum of y=sin(x)? __________________

    5. What is the minimum of y=sin(x)? __________________

    6. Find the maximum and minimum of y=cos(x)________________

    7. Find the maximum and minimum of y=tan(x)________________

    Exploration

    Click here for a JAVA applet to help you learn Transformation of trigonometric functions.

    1. Plot the following functions with your calculators:
      [Set your window to Xmin= -2π, Xmax= 2π, Xscl = p/2, Ymin= -3, Ymax= 3, and Yscl=0.5.]
      y1 = sin(x)
      y2 = sin(x) + 1.5
      y3 = sin(x) - 1.5
      1. How does the change in value c in f(x)+c affect the trigonometric function f(x)?
      2. Does the change in value c in f(x)+c affect both the amplitude and period of the trigonometric function f(x)?

    2. Now plot
      y1=cos(x),
      y2=cos(x-π/2),&
      y3=cos(x+π/3)
      1. How does the change in value d in f(x - d) affect the trigonometric function f(x)?
      2. Does the change in value d in f(x - d) affect both the amplitude and period of the trigonometric function f(x)?

    3. Now plot
      y1 = sin x
      y2= 2sin(x),
      y3= -sin x,
      y4 = (1/4)sin(x), &
      1. What is the amplitude of y= 2sin(x)?
      2. What is the amplitude of y= -sin(x)?
      3. What is the amplitude of y= (1/4)sin(x)?
      4. How does the change in value a in af(x) affect the trigonometric function f(x)?
      5. Does the change in value a affect the period and translation of the f(x)?
      6. How does negative a affect the trigonometric function f(x)?

    4. Now plot
      y1 = cos x
      y2 = cos (2x), &
      y3 = cos (0.5x).
      1. What is the period of y= cos(2x)?
      2. What is the period of y= cos(0.5x)?
      3. How does the change in value b in f(bx) affect the trigonometric function f(x)?
      4. Does the change in value b affect the amplitude and translation of the f(x)?

    5. Now plot
      y1 = cos(x) and
      y2 = cos(-x)
      What did you observe?

    6. Now plot
      y1 = cos(-x) and
      y2 = -cos(x).
      What did you observe?

    7. Plot y1 = tan(x), y2 = tan(-x) and y3 = - tan(x). What did you observe?

    8. Plot y1 = sin (x), y2 = sin(-x) and y3 = - sin(x). What did you observe?

Summary:


  • Study the function y = sin x above. Let us restrict for the moment to domain 0o ≤ x ≤ 360o .Observe that ymax = 1 when x=90o.
  • ymin = -1 when x=270o.
  • principal axis c = ymax + ymin
          2
    c = 0 when when x=0o, x=180o and x=360o.
  • The period = distance of three centers = 360o. Observe that in one period the curve forms a funny "s".
  • The amplitude is the distance from the center to the ymax or center to the ymin.
  • Amplitude is a positvie number. Amplitude= 1.

  • Study the function y = cos x above. Let us restrict ourselves for the moment to domain 0o ≤ x ≤ 360o .Observe that
    • ymax = 1 when x=0o.
    • ymin = -1 when x=180o.
    • principal axis c = ymax + ymin
            2
      c = 0 when x=90o and x=270o.
    • The period is the distance from peak to peak or trough to trough. Observe that in one period the curve forms a funny "v".
    • The amplitude is the distance from the center to ymax or ymin.
    • Amplitude is a positvie number. Amplitude= 1.
    Let Y1 = a sin (bx) + c . Study the sine curve Y1 and let us again restrict our domain to ONE period.
    • ymax = a [1] + c.
      This occurs when x=(1/4)period. Note maximum of [sin(bx)] is 1 and the minumum is -1.
    • ymin = a[-1] + c
      This occurs when x=(3/4)period.
    • principal axis c = ymax + ymin
            2
      c = 0 when x=(1/2)period and x=period.
    • Amplitude = |a| . Amplitude = (ymax-ymin)/2
    • Period = distance of 3 centers = 360o/b
    • IF Y1= -a sin(bx) + c then everything else as above but the curve here is a x-axis reflection of the above curve. Thus ymax= -a[-1] + c
      ymin= -a[1] + c

    Let Y2 = a cos (bx) + c . Study the cosine curve Y2 and let us again restrict our domain to ONE period.

    • ymax = a [1] + c.
      This occurs when x=(0)period and x=period. Note maximum of [sin(bx)] is 1 and the minumum is -1.
    • ymin = a[-1] + c
      This occurs when x=(1/2)period.
    • principal axis c = ymax + ymin
            2
      c = 0 when x=(1/4)period and x=(3/4)period.
    • Amplitude = | a | . Amplitude = (ymax-ymin)/2
    • Period = distance of peak to peak or trough to trough = 360o/b
    • IF Y2= -a cos(bx) + c then everything else as above but the curve here is a x-axis reflection of the above curve. Thus ymax= -a[-1] + c
      ymin= -a[1] + c

    Examples

    1. Find the amplitude and period of the following functions:
      1. y = 2sin(3x-4)
      2. y = 2-5sin(x)
      3. y = -(2/3)cos(0.5x)+6
      Solutions:
      1. y = af(bx -d) + c. The question is y = 2sin(3x-4). Thus, the amplitude is just a=2. The period is given by 2π/b. Thus, the period is 2π/3 or 1200.
      2. The question is y = 2-5sin(x). This can be rewritten as y = -5sin(x)+2. Thus, the amplitude is just |-5|=5. The negative sign in front of 5 affects reflection of the function and not the amplitude. Moreover, amplitude is a distance so a "-5" as the amplitude is not meaningful. The period is 2π/b. In this case, b=1. So the period is 2π or 3600.

      3. y = -(2/3)cos(0.5x)+6. The amplitude is 2/3. The period is 2π/b. In our case,b =0.5. Thus, the period is 2π/0.5 = 4π or 7200.

    2. Find the period and amplitude of y=2tan[(x/3)-4].
      Solutions:
      y = af(bx -d) + c. Here we may be tempted to say that the amplitude is a=2. But this is NOT correct. Plot this function out. In a tangent function we don't talk about amplitude because there is neither a maximum nor a minimum point in a tangent function. The period is however still π/b. In our case, the b is 1/3. Thus, the period is π/(1/3) = 3π.

    3. Find the maximum and minimum values of the following functions:
      1. y= 3sin(x)
      2. y= 2sin(x)-(1/2)
      3. y= 2sin(3x+π)
      4. y= -(2/3)cos[ (3x+π) /2]
      5. y= -(2/3)cos[ (3x+π) /2] + 1
      Solutions:
      To answer the questions regarding maximum and minimum for sine and cosine functions we only need to remember that the maximum and minimum for both sin(x) and cos(x) are 1 and -1 respectively.
      1. y=3sin(x). The function reaches its maximum when sin(x) is at the maximum. So maximum of y = 3[maximum of sin(x)] = 3(1). The maximum value is 3.
        The function reaches its minimum when sin(x) is at the minimum. So minimum of y=3[minimum of sin(x)]=3(-1). The minimum value is -3.
      2. y= 2sin(x)-(1/2). The function has its maximum value when sin(x) is maximum or sin(x)=1. The maximum of y is 2(1)-(1/2) = 3/2.
        The function reaches its minimum when sin(x) is minimum or sin(x)=-1. The minimum of y is 2(-1)-(1/2)=-5/2.
      3. y= 2sin(3x+π). Note the value 3 and π do not affect the amplitude nor vertical translation. Thus, we can treat our question the same way as we would with y=2sin(x). Thus, the maximum value is 2 and the minimum value is -2 as in example (i) above.
      4. y= -(2/3)cos[ (3x+π) /2]. Same argument as example (iii) above. We are basically working with a problem y=-(2/3)cos(x). The function reaches its maximum when cos(x) is at its minimum or cos(x)=-1. So the maximum value is -(2/3)(-1) = 2/3.
        The function reaches its minimum when cos(x) is at its maximum or cos(x)=1. The minimum is -(2/3)(1) = -2/3.
      5. y= -(2/3)cos[ (3x+π) /2] + 1. Here is a slight modification of the question (iv) above. The value +1 translates the whole function a unit vertically upward. Thus, the maximum value is 2/3+1 = 5/3 and the minimum value is -2/3+1 = 1/3.
      You should confirm all the given solutions with graphs.

    4. The population of a species of insect in a garden can be modelled by the function:
      P = 500 + 200sin(πT/6), 0≤ T ≤ 12
      where T is measured in weeks after the initial population estimate.
      1. What is the initial population?
      2. What is the largest population?
      3. When will the largest population be achieved?
      4. When does the population reach 600?
      Solutions:
      1. The initial population is simply 500 when t=0 so sin(0)=0.
      2. The largest population is when sin(x)=1. Thus, largest population is 500 + 200(1)= 700.
      3. sin(x) =1
        sin (πT/6) = 1
        πT/6 = sin-1 1
        πT/6 = π/2
        T/6 = 1/2
        T = 6/2
        T = 3. [The third week]
        Is there any other answer? Here we need to consider the period. The period of this function is 2π/(π/6) = 12. Thus, the next peak will be at 15 but 15 is not part of our domain. So there is a unique solution.We can also confirm this fact with a graph.
      500 + 200sin(πT/6) = 600
      200sin(πT/6) = 600-500
      200sin(πT/6) = 100
      sin(πT/6) = 1/2
      (πT/6) = π/6
      T = 1
      Use a graph and we will quickly realize that there are two answers. 600 is before the peak population of 700. From T=1 to the peak time (T=3) there is a 2 week duration. The function is symmetric thus the next time the population reaches 600 must be T=3+2 or the fifth week.
      Answers: First and fifth week. Confirm these with a graph.

    Summary:
    Let f(x) be a trigonometric function that could be sine, cosine or tangent.
    y = a f(bx - d) + c
    • A change in c translates the (whole) function vertically by c units. If c is positive then the translation is vertically upward. If the c is negative then the translation is vertically downward.
    • A change in d translates the (whole) function horizontally by d units. If the d is positive then the translation is horizontally to the left. If the d is negative then the translation is horizontally to the right.
    • A change in a affects the amplitude of the function. If |a| > 1 then the amplitude is amplified (increased) to a units [vertical (positive) dilation]. If |a| < 1 then the amplitude is decreased to a units [vertical (negative) dilation]. Note that this generation does not apply to f(bx-d) = tan(bx-d) because there is no amplitude to speak of in a tangent function.
    • If a carries the negative sign then the function is reflected in the x-axis.
    • A change in b affects the period of the function to 2π/b . If |b|>1 then the function is horizontally compressed. If |b|<1 then the function is horizontally dilated. However, if f(bx-d)= tan(bx-d) then the period is π/b.
    • Note sin(-x) = -sin(x) and tan(-x) = -tan(x).
    • cos(x) = cos(-x) but cos(-x) does NOT equal -cos(x).
    Other Resources:
  • http://www.teachers.ash.org.au/mikemath/algtrigmodel/ This is a good collection of real-life models with trigonometric functions. Contains also links to exercises with solutions.
  • http://www.niwa.cri.nz/edu/resources/climate/modelling/ A climate model exercise with sin and cosine function. Use real data.
  • http://people.hofstra.edu/faculty/Stefan_Waner/trig/trig1.html A tutorial on trigonometric function with some exercises.
  • http://www.travel.com.hk/weather/china.htm Contains some whether data on some major cities in China.
  • http://www.info.gov.hk/censtatd/eng/hkstat/ Contains statistical data of Hong Kong. Some can be modelled using trigonometric functions.