📈 Level 2 - Topic 2: Phase Shift, Amplitude, and Period of Trigonometric Functions 📊📐

1) Transforming Trigonometric Graphs - Beyond the Basics 🚀

In the previous topic, we explored the basic graphs of \( y = \sin(\theta) \), \( y = \cos(\theta) \), and \( y = \tan(\theta) \). Now, we'll learn how to transform these basic graphs to create a wider variety of trigonometric functions. These transformations are essential for modeling real-world periodic phenomena with greater precision.

Why Transform Graphs?

  • Flexibility in Modeling: Basic sine and cosine waves have fixed amplitude and period. Transformations allow us to adjust these to model phenomena with different intensities, frequencies, and starting points.
  • Understanding Complex Waves: Many real-world waves (sound, light, oscillations) are not simple sine or cosine waves. Transformations help us analyze and synthesize more complex wave patterns.
  • Graphical Analysis: Understanding transformations enhances our ability to interpret trigonometric graphs and extract meaningful information from them.

We'll focus on three primary types of transformations: Phase Shift, Amplitude Change, and Period Change. We'll also briefly touch upon Vertical Shift.

2) Amplitude - Vertical Stretch and Compression ↕️

Amplitude affects the vertical stretch or compression of a trigonometric graph. Consider the general forms:

\( y = A \sin(\theta) \) and \( y = A \cos(\theta) \)

Here, \( |A| \) represents the amplitude.

Amplitude (\( |A| \)): The maximum vertical displacement of the graph from its midline (horizontal axis for \( y = A \sin(\theta) \) and \( y = A \cos(\theta) \)). It's always a non-negative value. If \( A \) is negative, it also reflects the graph across the \( \theta \)-axis (more on reflection later).

Effect of Amplitude \( |A| \):

  • If \( |A| > 1 \): Vertical stretch - the graph is stretched vertically, making it taller. The maximum and minimum values become \( A \) and \( -A \) respectively.
  • If \( 0 < |A| < 1 \): Vertical compression - the graph is compressed vertically, making it shorter. The maximum and minimum values are still \( A \) and \( -A \), but closer to the \( \theta \)-axis.
  • If \( A < 0 \): Vertical stretch/compression combined with reflection across the \( \theta \)-axis. For example, \( y = -\sin(\theta) \) is the sine graph reflected upside down. The amplitude is still \( |-1| = 1 \).

Example 1: Amplitude Change

Compare the graphs of \( y = \sin(\theta) \), \( y = 3\sin(\theta) \), and \( y = 0.5\sin(\theta) \).

  1. \( y = \sin(\theta) \) has amplitude 1 (standard sine wave).
  2. \( y = 3\sin(\theta) \) has amplitude 3 - vertically stretched by a factor of 3. Range is \( [-3, 3] \).
  3. \( y = 0.5\sin(\theta) \) has amplitude 0.5 - vertically compressed by a factor of 0.5. Range is \( [-0.5, 0.5] \).

View Graph of Amplitude Changes - Opens in new tab

*(Link will open an illustrative graph in WolframAlpha. Feel free to replace with your preferred graph.)*

3) Period - Horizontal Stretch and Compression ↔️

Period affects the horizontal stretch or compression of a trigonometric graph, changing how often the function completes a cycle. Consider the general forms:

\( y = \sin(B\theta) \) and \( y = \cos(B\theta) \)

Here, \( B \) affects the period.

Period (with \( B \)): For functions \( y = \sin(B\theta) \) and \( y = \cos(B\theta) \), the period is given by:

Period \( = \frac{360^\circ}{|B|} \) (in degrees) or Period \( = \frac{2\pi}{|B|} \) (in radians)

\( |B| \) is always taken as a positive value. \( B \) effectively compresses or stretches the graph horizontally.

Effect of \( B \):

  • If \( |B| > 1 \): Horizontal compression - the graph is compressed horizontally, completing cycles more quickly. The period becomes shorter.
  • If \( 0 < |B| < 1 \): Horizontal stretch - the graph is stretched horizontally, completing cycles more slowly. The period becomes longer.
  • If \( B < 0 \): Horizontal stretch/compression combined with reflection across the y-axis. However, since \( \sin(-\theta) = -\sin(\theta) \) and \( \cos(-\theta) = \cos(\theta) \), the reflection for cosine is visually indistinguishable, and for sine, it can be combined with vertical reflection due to amplitude. We usually consider \( B > 0 \) for period changes and handle reflections separately if needed.

Example 2: Period Change

Compare the graphs of \( y = \cos(\theta) \), \( y = \cos(2\theta) \), and \( y = \cos(0.5\theta) \).

  1. \( y = \cos(\theta) \) has period \( 360^\circ \) (standard cosine wave).
  2. \( y = \cos(2\theta) \) has period \( \frac{360^\circ}{2} = 180^\circ \) - horizontally compressed, completes cycles twice as fast.
  3. \( y = \cos(0.5\theta) \) has period \( \frac{360^\circ}{0.5} = 720^\circ \) - horizontally stretched, completes cycles half as fast.

View Graph of Period Changes - Opens in new tab

*(Link will open an illustrative graph in WolframAlpha. Feel free to replace with your preferred graph.)*

Frequency: Period and frequency are inversely related. Frequency ( \( f \)) is the number of cycles per unit of \( \theta \) (e.g., cycles per \( 360^\circ \) or \( 2\pi \) radians). If Period \( = T \), then Frequency \( = f = \frac{1}{T} \). A shorter period means higher frequency, and a longer period means lower frequency. \( |B| \) is directly related to the frequency change relative to the basic function.

4) Phase Shift - Horizontal Translation ↔️➡️⬅️

Phase Shift causes a horizontal translation (shift) of the trigonometric graph, moving it left or right along the \( \theta \)-axis. Consider the general forms:

\( y = \sin(\theta - C) \) and \( y = \cos(\theta - C) \)

Here, \( C \) represents the phase shift.

Phase Shift (\( C \)): A horizontal shift of the graph.

  • If \( C > 0 \): Shift to the right by \( C \) units.
  • If \( C < 0 \): Shift to the left by \( |C| \) units.
It indicates how much the graph is shifted horizontally compared to the basic sine or cosine graph.

Example 3: Phase Shift

Compare the graphs of \( y = \sin(\theta) \), \( y = \sin(\theta - 45^\circ) \), and \( y = \sin(\theta + 30^\circ) \).

  1. \( y = \sin(\theta) \) is the standard sine wave (no phase shift).
  2. \( y = \sin(\theta - 45^\circ) \) is shifted \( 45^\circ \) to the right.
  3. \( y = \sin(\theta + 30^\circ) = \sin(\theta - (-30^\circ)) \) is shifted \( 30^\circ \) to the left.

View Graph of Phase Shifts - Opens in new tab

*(Link will open an illustrative graph in WolframAlpha. Note: Radians are used in the link for phase shift values as fractions of pi. Feel free to replace with your preferred graph, possibly in degrees.)*

5) Vertical Shift - Vertical Translation ⬆️⬇️

Although not in the topic title, Vertical Shift is another important transformation. It causes a vertical translation (shift) of the graph, moving it up or down along the y-axis. Consider the general forms:

\( y = \sin(\theta) + D \) and \( y = \cos(\theta) + D \)

Here, \( D \) represents the vertical shift.

Vertical Shift (\( D \)): A vertical shift of the graph. The entire graph is moved:

  • If \( D > 0 \): Shift upwards by \( D \) units.
  • If \( D < 0 \): Shift downwards by \( |D| \) units.
The midline of the graph (horizontal axis for basic sine/cosine) shifts to \( y = D \).

Example 4: Vertical Shift

Compare the graphs of \( y = \cos(\theta) \), \( y = \cos(\theta) + 2 \), and \( y = \cos(\theta) - 1 \).

  1. \( y = \cos(\theta) \) is the standard cosine wave (no vertical shift, midline at \( y=0 \)).
  2. \( y = \cos(\theta) + 2 \) is shifted 2 units upwards. Midline at \( y = 2 \).
  3. \( y = \cos(\theta) - 1 \) is shifted 1 unit downwards. Midline at \( y = -1 \).

View Graph of Vertical Shifts - Opens in new tab

*(Link will open an illustrative graph in WolframAlpha. Feel free to replace with your preferred graph.)*

6) Combining Transformations - The General Form 🎨

We can combine all these transformations to get the general forms of transformed sine and cosine functions:

\( y = A \sin(B(\theta - C)) + D \) and \( y = A \cos(B(\theta - C)) + D \)

In these general forms:
  • \( A \) affects Amplitude (Amplitude \( = |A| \)).
  • \( B \) affects Period (Period \( = \frac{360^\circ}{|B|} \) or \( \frac{2\pi}{|B|} \)).
  • \( C \) affects Phase Shift (Horizontal shift by \( C \) units; right if \( C > 0 \), left if \( C < 0 \)).
  • \( D \) affects Vertical Shift (Vertical shift by \( D \) units; upwards if \( D > 0 \), downwards if \( D < 0 \)).

Example 5: Combined Transformations

Analyze the function \( y = 2\sin(3(\theta - 30^\circ)) + 1 \).

  1. Amplitude: \( A = 2 \), so Amplitude \( = |2| = 2 \) (vertically stretched by a factor of 2).
  2. Period: \( B = 3 \), so Period \( = \frac{360^\circ}{3} = 120^\circ \) (horizontally compressed, period is shorter).
  3. Phase Shift: \( C = 30^\circ \), so Phase Shift \( = 30^\circ \) to the right.
  4. Vertical Shift: \( D = 1 \), so Vertical Shift \( = 1 \) unit upwards (midline is \( y = 1 \)).

View Graph of Combined Transformations - Opens in new tab

*(Link will open an illustrative graph in WolframAlpha. Feel free to replace with your preferred graph.)*

7) Practice Questions 🎯

7.1 Fundamental – Identifying Transformations

1. For the function \( y = 4\cos(\theta) \), identify the amplitude and period.

2. For the function \( y = \sin(0.5\theta) \), identify the amplitude and period.

3. For the function \( y = \sin(\theta - 60^\circ) \), identify the phase shift.

4. For the function \( y = \cos(\theta) + 3 \), identify the vertical shift.

5. For the function \( y = -2\sin(\theta) \), identify the amplitude and reflection.

6. What is the period of \( y = \sin(3\theta) \) in degrees and radians?

7. What is the phase shift of \( y = \cos(\theta + 90^\circ) \)? Is it a shift to the left or right?

8. What is the amplitude of \( y = 0.8\sin(\theta) \)? Is it a vertical stretch or compression?

9. Describe the transformation needed to change \( y = \cos(\theta) \) into \( y = \cos(\theta - 180^\circ) \).

10. True or False: Amplitude always changes the period of a trigonometric function. Explain.

7.2 Challenging – Analyzing Combined Transformations & Equations 💪🚀

1. Analyze the function \( y = -3\cos(2(\theta + 45^\circ)) - 2 \). Identify the amplitude, period, phase shift, vertical shift, and reflection.

2. Write the equation of a sine function that has an amplitude of 5, a period of \( 90^\circ \), and a phase shift of \( 30^\circ \) to the right, and is shifted vertically upwards by 4 units.

3. How does the graph of \( y = \sin(\theta) \) compare to the graph of \( y = \sin(\theta + 360^\circ) \)? Explain in terms of phase shift and period.

4. If a cosine function has a period of \( 120^\circ \), what is the value of \( B \) in the form \( y = \cos(B\theta) \)?

5. (Conceptual) Explain how you could visually determine the amplitude, period, phase shift, and vertical shift of a given sinusoidal graph. What key points or features on the graph would you look for to identify each transformation?

8) Summary - Mastering Trigonometric Graph Transformations 🎉

  • Transformations: Amplitude, Period, Phase Shift, and Vertical Shift are key transformations of trigonometric graphs.
  • Amplitude \( |A| \): Vertical stretch/compression. Amplitude is \( |A| \). Affects the range.
  • Period (with \( B \)): Horizontal stretch/compression. Period \( = \frac{360^\circ}{|B|} \) or \( \frac{2\pi}{|B|} \). Affects the frequency.
  • Phase Shift \( C \): Horizontal translation. \( y = f(\theta - C) \) shifts right if \( C > 0 \), left if \( C < 0 \).
  • Vertical Shift \( D \): Vertical translation. \( y = f(\theta) + D \) shifts up if \( D > 0 \), down if \( D < 0 \). Affects the midline.
  • General Forms: \( y = A \sin(B(\theta - C)) + D \) and \( y = A \cos(B(\theta - C)) + D \) combine all transformations.
  • Analyzing and Constructing Graphs: Understanding these transformations allows us to analyze given transformed graphs and construct equations for graphs with desired properties.

Congratulations! You've now mastered the fundamental transformations of trigonometric graphs! Understanding amplitude, period, phase shift, and vertical shift opens up a powerful toolkit for analyzing and modeling periodic phenomena. In the next topics, we'll explore more advanced trigonometric functions, inverse functions, and identities, building upon this graphical understanding. Keep practicing and you'll become a trigonometry graph transformation expert! 📈📊📐🌟

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