Sinusoidal Function Transformations Study Notes

Topic 3.6: Sinusoidal Function Transformations

Introduction to Sinusoidal Functions

  • We previously learned how to identify characteristic properties from the graph of a sinusoidal function in Topics 3.4 – 3.5.
  • The characteristics studied include:
      - Midline
      - Amplitude
      - Period
  • This section extends those concepts to understand how these properties influence the equations of sinusoidal functions through transformations.

Transformations of Sinusoidal Functions

  • The properties of a sinusoidal function affect its graph through the following transformations:
      1. Vertical Dilation:
         - A vertical dilation occurs by a factor of aa, where aa defines the amplitude of the graph.
      2. Vertical Translation:
         - A vertical translation of dd units occurs, where dd represents the midline of the graph.
      3. Horizontal Dilation:
         - A horizontal dilation occurs by a factor of bb, where the period PP is given by P=2extπbP = \frac{2 ext{π}}{b}.
  • Thus, a function can be represented as:
    y=aextsin(b(x−c))+dy = a ext{sin}(b(x - c)) + d
      where
      - aa = amplitude
      - bb = frequency related to the period
      - dd = midline / vertical shift
      - cc = phase shift

Examples of Sinusoidal Functions

Example 1
  • Function: h(x)h(x)
  • Given properties:
      - Amplitude: a=6a = 6
      - Period: P=2extπb=extπP = \frac{2 ext{π}}{b} = ext{π}
      - Therefore, b=2b = 2
      - Vertical translation: d=3d = 3
Example 2
  • Function: f(heta)f( heta)
  • Properties:
      - Midline: y=3y = 3 (vertical translation d=3d = 3)
      - Period: P=2extπb=extπP = \frac{2 ext{π}}{b} = ext{π}, thus b=2b = 2
      - Amplitude: a=3a = 3
  • Expression: f(heta)=3extsin(2heta)+3f( heta) = 3 ext{sin}(2 heta) + 3
Example 3
  • Evaluate the properties of a sinusoidal function gg:
      - Choices:
        - (A) Period: extπ,ext{π}, Amplitude: 22
        - (B) Period: extπ,ext{π}, Amplitude: 44
        - (C) Period: 2extπ,2 ext{π}, Amplitude: 22
        - (D) Period: 2extπ,2 ext{π}, Amplitude: 44
      - Need to determine the values based on the graph presented.
Example 4
  • Given a function kk with a maximum point at (0, 6) and the next minimum at π2\frac{π}{2}:
      - Period is extπext{π}, so rac{2 ext{π}}{b} = ext{π}
    ightarrow b = 2
      - Midline: 6+−42=16 + \frac{-4}{2} = 1
      - Amplitude: 6−1=56 - 1 = 5
      - Analyze provided options for valid expressions for k(x)k(x).
Example 5
  • Function ff has:
      - Amplitude: 33
      - Midline: y=−1y = -1
      - Period: P = ext{π}
    ightarrow b = 2
  • Evaluate potential graph candidates based on these parameters.
Example 6
  • The graph presents a function with:
      - Midline: y=−1y = -1
      - Amplitude: 33
      - Period: 2extπ2 ext{π}
      - Analyze various expressions to determine correctness based on these parameters.
  • The cosine option reflected over the midline appears to be correct.
Phase Shift of a Sinusoidal Function
  • Definition: A phase shift is a horizontal translation of the graph of a sinusoidal function.
  • A function can be translated horizontally as follows:
      - For h(x)=aextsin(b(x−c))+dh(x) = a ext{sin}(b(x - c)) + d, the graph is a phase shift of the sine graph by cc units to the right.
  • This transformation equally applies to the cosine function:
      - k(x)=aextcos(b(x−c))+dk(x) = a ext{cos}(b(x - c)) + d
Example 7
  • Given a graph with a midline of y=1y = 1, amplitude of 22, and period of extπ,ext{π}, analyze the choices:
      - (A) through (D) provide distinctive relationships based on the given midline and amplitude. Evaluate each option carefully for correctness based on the sine and cosine characteristics.
Example 8
  • The graph represents two full cycles, with labeled points indicating characteristics of the function.
  • Constants:
      - Midline: y = 11
    ightarrow d = 11
      - Amplitude: 5
    ightarrow a = 5
      - Period: 12
    ightarrow rac{2 ext{π}}{b} = 12
    ightarrow b = rac{ ext{π}}{6}
  • The sine curve's maximum occurs based on transformations where:
    b(t+c)=kb(t + c) = k (focusing particularly where tt meets known parameters)

Conclusion

  • Understanding the transformations specific to sinusoidal functions is crucial for analyzing their equations and graphs. Each transformation (vertical dilation, vertical translation, horizontal dilation, and phase shift) affects the overall graph in specific and predictable ways.
  • Careful analysis of given functions and their parameters aids in identifying valid expressions related to those functions.

Important Notes

  • Ensure to capture both the analysis of graphs and the deductions of parameters correctly from visual cues.
  • Practice with different sinusoidal functions to solidify the application of these transformations in various contexts.