State the phase shift for the graph. The given task is to graph one complete cycle for the function:\[ y = 4 - 3 \tan\left(\frac{1}{3}x + \frac{\pi}{6}\right) \]### Description of the GraphsThe diagram consists of multiple graphs showing the transformation of a tangent function. Each subplot is trying to depict a segment of the function over a specified range.#### Axes and Labels- The horizontal axis (x-axis) ranges from $-\frac{3}{4}\pi$ to $3\pi$.- The vertical axis (y-axis) ranges approximately from $-30$ to $30$.#### Graph Details- Each graph shows segments of a transformed tangent function.- Asymptotes, typical for tangent functions, are evident where the function's value tends toward positive or negative infinity.- The function appears to have vertical asymptotes where the graph is not defined.#### Function Description- The basic tangent function has been shifted and stretched: - **Vertical Shift**: The "+4" shifts the graph up by 4 units. - **Amplitude**: The multiplication by "−3" flips the graph vertically and stretches it. - **Horizontal Shift**: The argument $\frac{1}{3}x + \frac{\pi}{6}$ involves a phase shift and stretching due to $\frac{1}{3}$. - The cycle of the transformation affects the angle and the position of the vertical asymptotes.#### Evaluation- The correct graph accurately represents the transformed tangent function over its complete cycle, from $-\frac{3}{4}\pi$ to $3\pi$.- The check mark indicates which of the depicted functions represents this transformation correctly.### ConclusionUnderstanding these transformations is critical for mastering trigonometric functions and their graphical representations. This exercise demonstrates the impact of parameter changes on the tangent function's shape and position.

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Answered: Graph one complete cycle for the…

Math

Trigonometry

Graph one complete cycle for the following function. y%3 4-3 tan(r + 등)

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

State the phase shift for the graph.

The given task is to graph one complete cycle for the function:

\[ y = 4 - 3 \tan\left(\frac{1}{3}x + \frac{\pi}{6}\right) \]

### Description of the Graphs

The diagram consists of multiple graphs showing the transformation of a tangent function. Each subplot is trying to depict a segment of the function over a specified range.

#### Axes and Labels

- The horizontal axis (x-axis) ranges from $-\frac{3}{4}\pi$ to $3\pi$.
- The vertical axis (y-axis) ranges approximately from $-30$ to $30$.

#### Graph Details

- Each graph shows segments of a transformed tangent function.
- Asymptotes, typical for tangent functions, are evident where the function's value tends toward positive or negative infinity.
- The function appears to have vertical asymptotes where the graph is not defined.

#### Function Description

- The basic tangent function has been shifted and stretched:
- **Vertical Shift**: The "+4" shifts the graph up by 4 units.
- **Amplitude**: The multiplication by "−3" flips the graph vertically and stretches it.
- **Horizontal Shift**: The argument $\frac{1}{3}x + \frac{\pi}{6}$ involves a phase shift and stretching due to $\frac{1}{3}$.
- The cycle of the transformation affects the angle and the position of the vertical asymptotes.

#### Evaluation

- The correct graph accurately represents the transformed tangent function over its complete cycle, from $-\frac{3}{4}\pi$ to $3\pi$.
- The check mark indicates which of the depicted functions represents this transformation correctly.

### Conclusion

Understanding these transformations is critical for mastering trigonometric functions and their graphical representations. This exercise demonstrates the impact of parameter changes on the tangent function's shape and position.

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