Math127 Graph Of Sine & Cosine Functions

Introduction to Sine and Cosine Functions

• Focus on intersection between sine and cosine graphs

• Domain: (-∞, +∞)

• Range: [-1, 1]

  • Highest point: 1

  • Lowest point: -1

  • Exceptions due to vertical shifts

Symmetry of Graphs

• Sine Function:

  • Exhibits origin symmetry

  • Starts from (0,0)

  • Symmetry in Quadrants 1 and 3

• Cosine Function:

  • Exhibits y-axis symmetry

  • Starts from highest point (1)

Graphing Sine and Cosine Functions

Key Points Acquisition

• Horizontal shifting: (\frac{\pi}{2}) to determine five key points

• Five key points indicate the starting and changing points of the graph

• Calculation: Period divided by 4 for key intervals

Important Formulas for Graphing

• Amplitude: |a| (absolute value of coefficient in front of sine or cosine)

• Period: (\frac{2\pi}{b}) (where b is the coefficient of x)

• Horizontal shifting: (\frac{c}{b}) (horizontal shift value)

• Vertical shift: d (constant added)

• Find five key points by adding period/4 to starting point

Example 1: Graphing (y = \frac{1}{2} \sin x)

• Amplitude: |(\frac{1}{2})| = (\frac{1}{2})

• Period: (2\pi)

• Key interval: (\frac{2\pi}{4} = \frac{\pi}{2})

• Five key points:

  • 0, (\frac{\pi}{2}), (\pi), (\frac{3\pi}{2}), and (2\pi)

• Graph starts at (0), then rises to (\frac{1}{2}) then down through graph with smooth curves

Example 2: Different Amplitude (y = -2 \sin x)

• Amplitude: |(-2)| = 2

• Period: (2\pi)

• Key interval: (\frac{\pi}{2})

• Start from (0), go down first (due to negative), and rise using the five key points repeated from Example 1

Adjusting the Value of b

Example: Graphing with (b)

• If gaus (y = 3 \sin(2x))

• Amplitude: |3| = 3

• Period is (\frac{2\pi}{2} = \pi)

• Key interval: (\frac{\pi}{4})

• Five key points generated

• Add (\frac{\pi}{4}) sequentially for key points in graph

Example with Horizontal Shifting (y = 3 \sin(2x + \pi))

• Amplitude: |3| = 3

• Period: (\pi)

• Horizontal shifting = (-\frac{\pi}{4})

• Starting point adjusted to account for horizontal shift

• Graph creation follows traditional process, adjusting points accordingly

Example with Cosine Function

• Cosine example: (y = -3 \cos x)

• Amplitude: |(-3)| = 3

• Period: (2\pi)

• Begin from lowest point due to negative value

• Create five key points and observe the graph ascend and descend based on amplitude

Final Examples Combining Values (y = \frac{1}{2} \cos(4x + \pi))

• Amplitude: |(\frac{1}{2})| = (\frac{1}{2})

• Period: (\frac{\pi}{2})

• Horizontal shifting: (-\frac{\pi}{4})

• Five key points calculated and plotted accordingly

Summary

• Understanding of sine and cosine functions enables correct graphing

• Key characteristics such as amplitude, period, and symmetry aid in visual representation

• Practice is necessary to master identifying and graphing these functions with adjustments.