(90) Math12550 Exam3 Review Part 2 of 6: Graphs of Trigonometric Functions

Overview of Trigonometric Functions

• Trigonometric functions relate angles to ratios and have a graphical representation.

Example 1: Graphing a Cosine Function

• Function to graph: ( f(x) = -2 \cos(4x) + 2 )

• Involves multiple transformations of the cosine function.

Starting with the Base Function

• Begin with the graph of ( \cos(x) ):

  • Oscillates between 1 and -1.

  • Has a period of ( 2\pi ).

Transforming the Cosine Function

• Find ( \cos(4x) ):

  • The period is given by ( \frac{2\pi}{b} ), where ( b = 4 ).

  • Therefore, the period of ( \cos(4x) ) is ( \frac{2\pi}{4} = \frac{\pi}{2} ).

  • It completes four periods in the interval ( [0, 2\pi] ).

Vertical Scaling and Reflection

• Scaling by -2:

  • Flips the graph of ( \cos(4x) ) across the x-axis.

  • Oscillation range changes from between -1 and 1 to between -2 and 2, maintaining the period of ( \frac{\pi}{2} ).

Vertical Shift

• Add a vertical shift of +2:

  • Final function oscillates between 0 and 4.

  • Midline shifts to ( y = 2 ).

Summary of Final Function

• Final function is: ( f(x) = -2 \cos(4x) + 2 )

• Key features:

  • Amplitude: 2

  • Period: ( \frac{\pi}{2} )

  • Midline: ( y = 2 )

Example 2: Identifying a Trigonometric Function from a Graph

• Graph shown; need to identify the function.

Observing Key Characteristics

• Look for:

  • Amplitude: 1

  • Midline: ( y = 0 )

  • Period: 2\pi;

  • Crest of the wave starts at ( -\frac{\pi}{4} ) and ends at ( \frac{7\pi}{4} ).

Determining Possible Functions

• Can be modeled as either:

  • Cosine function: ( \cos(x + \frac{\pi}{4}) )

    • Phase shift: ( \frac{\pi}{4} ) to the left.

  • Sine function: ( \sin(x + \frac{3\pi}{4}) )

    • Phase shift: ( \frac{3\pi}{4} ) to the left.

Example 3: Another Graph Identification

• Analyzing another graph, focus again on key features:

Extracting Information from the Function

• Amplitude: 3

• Midline: ( y = 0 )

• Determine period:

  • Full rotation observed, period is 2.

Final Function Formulation

• Initially, assume a sine function: ( 3 \sin(a \cdot x) )

• Relate period with the coefficient: ( \text{If period is 2, then } \frac{2\pi}{b} = 2. )

• Solve for b:

  • ( b = \pi. )

• Function modeled as ( 3 \sin(\frac{\pi}{2} x) ) or other suitable transformations satisfied.