Graphing Transformations of Trig Fumctions

Transformations of Sine and Cosine Functions

Key Points of the Cosine Function:

• Basic cosine function: (y = \cos(x)) has key points: (0, 1), (\frac{\pi}{2}, 0), (\pi, -1), (\frac{3\pi}{2}, 0), (2\pi, 1).

Mapping Notation:

• Amplitude (A): Distance from the midline. Positive = upward.

• Vertical Translation (D): Shifts graph up/down (e.g., D = 6 moves up).

• Phase Shift (C): Horizontal movement. Positive right, negative left.

• Period (P): (P = \frac{2\pi}{B}), where B is frequency.

Example Function:

• y = 5 \cos\left(2x - \frac{\pi}{2}\right) + 6

Graphing Scale Calculation:

• Divide period into four segments. For P = \pi, each segment = \frac{\pi}{4}.

Mapping Transformed Coordinates:

• X Mapping: X' = \frac{X}{2} - \frac{\pi}{2}

• Y Mapping: Y' = -5Y + 6

Example Calculations:

• For (0, 1): New point = (-\frac{\pi}{2}, 1)

• For (\frac{\pi}{2}, 0): New point = (0, 6)

• For (\pi, -1): New point = (0, 11)

Final Note on Plotting:

• Plot new points like ((-\frac{\pi}{2}, 1), (0, 6), (0, 11),...) to visualize the graph while maintaining cosine shape.