Summary of How To Graph Trigonometric Functions | Trigonometry

Main Ideas and Concepts

• Graphing Trigonometric Functions:
  • Focuses on sine and cosine functions and their transformations.
  • Each function has a distinct shape, with sine starting at the origin and cosine starting at the maximum.
• Understanding the Sine Function:
  • The Sine Function is sinusoidal, with a period of \(2\pi\).
  • A negative Sine Function flips the graph over the x-axis.
• Understanding the Cosine Function:
  • The Cosine Function also has a period of \(2\pi\), but starts at its maximum value.
  • A negative Cosine Function flips the graph over the x-axis.
• Key Points for Graphing:
  • Each cycle (period) can be broken down into four key points: \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
  • For two cycles, extend this to \(4\pi\) and identify additional key points.
• Amplitude:
  • The Amplitude (the height of the wave) is determined by the coefficient in front of the sine or Cosine Function.
  • It indicates the vertical stretch or compression of the graph.
• Period Calculation:
  • The period of a function can be found using the formula \( \text{Period} = \frac{2\pi}{b} \), where \(b\) is the coefficient of \(x\) in the function.
• Vertical Shifts:
  • A vertical shift is applied by adding or subtracting a constant to/from the function, affecting the midline of the graph.
• Phase Shift:
  • Phase shifts occur when there is a constant added to the \(x\) term inside the function. This shifts the graph left or right.
  • The Phase Shift can be calculated by setting the inside of the function equal to zero.

Methodology for Graphing Trigonometric Functions

• For Sine and Cosine Functions:
  • Identify the Amplitude and period.
  • Break the period into four key points.
  • Plot the key points and draw the wave shape.
  • For negative functions, flip the graph over the x-axis.
  • If there’s a vertical shift, adjust the midline accordingly.
  • For phase shifts, determine where the wave starts on the x-axis.
• Example Steps:
  • Graphing \(y = 2\sin(x)\):
    • Amplitude: 2
    • Period: \(2\pi\)
    • Key points: \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\)
    • Plot points and shape based on Amplitude.
  • Graphing \(y = -3\cos(x)\):
    • Amplitude: 3
    • Start at the bottom due to the negative sign.
    • Adjust for Vertical Shifts if applicable.

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