4.1-4.2 Lecture

Amplitude: The height of the wave from the midline to the peak (or trough). Example: For a function represented as a sine or cosine, if the amplitude is 3, the maximum value (peak) would be midline value + amplitude, and the minimum value (trough) would be midline value - amplitude.
Midline: The horizontal line that represents the central value of the graph. Example: For a function such as y = 4, the midline is at y = 4.
Period: The distance over which the function completes one cycle. Calculated as Period = (2 * π) / b, where b is the coefficient of x in the function.

Graphing Functions
Example function: Consider f(x) = -cos(bx) + c.

Graphing Steps

  1. Identify amplitude, midline, and period.

    • Amplitude: In the example, if the function is f(x) = -cos(x) where the amplitude is 1, the maximum is at the midline + amplitude and minimum at midline - amplitude.

    • Period: If b = 1, then Period = 2 * π.

  2. Create a rough sketch of the midline:

    • If the choices in a graph are 1 to 7 (e.g., y = 4), plot the midline on that number.

  3. Plot the maximum and minimum values — Figuring out where the function reaches its peaks and troughs with reference to midline and amplitude:

    • Midline at 4, amplitude at 3, then maximum value is 4 + 3 = 7, and the minimum value is 4 - 3 = 1.

  4. Define the zero crossings, where the function intersects the midline.

  5. Note the starting point for cosine versus sine functions. A cosine function starts at its maximum while a sine function starts at the midline.

Example: Graph of f(x) = 2 * sin(6x) + 3

  • Period: Calculated as Period = (2 * π) / 6 = π / 3.

  • Amplitude: 2

  • Midline: y = 3

Visual Representation

  1. Midline at 3

  2. From the midline calculate max: 3 + 2 = 5, and min: 3 - 2 = 1.

  3. Identify period to enhance the x-axis (two full periods would go to (2 * π) / 3 * 2 = (4 * π) / 3).

  4. Divide into quarter periods for accurate graphing.

  5. Label Points:

    • Quarter points would be at: π / 12, 3 * π / 12, 5 * π / 12, etc.

Important Considerations

  • Graphing should cover two full cycles.

  • Check calculations through reference angles and unit circle.

  • When calculating distances between points, verify that differences reflect the period accurately.

Horizontal Shifts and Reflections

  • Shifting a graph (e.g., left or right) depends on the number inside the function. A negative number indicates a shift to the right.

  • Reflection is done by multiplying by negative, such as -sin(x) starting from midline going downwards.

Complex Transformations

  • When transformations combine multiple effects (horizontal stretch/compression, reflections, and shifts), calculate the period first then apply corrections:

    • Example: g(x) = -sin(3x - π) involves stretching and shifting.

    • Handle b first in terms of period, then apply any horizontal shifts as adjustments to the graph at the end of plotting before connecting dots.

Example: Complex Transformation Graphing

  • If the equation is g(x) = -sin(3x + 1)

    1. Period: Period = (2 * π) / 3.

    2. Identify Midline: Assume no vertical shifts thus at 0.

    3. Amplitude: 1.

    4. Apply shifts last while graphing to avoid confusion in process. Add cosine trends where appropriate.