Question is attached The image contains a mathematical expression and a question related to it. Here is the transcription:Expression:\[ Y = -8 \cos \left(6x + \frac{2\pi}{3}\right) + 4 \]Question:"Is there a vertical expansion or compression? Pick one and explain why."---### ExplanationThe expression given is a transformation of the cosine function. - **Vertical Expansion/Compression**: - The coefficient of the cosine function, $-8$, indicates a vertical transformation. Since $|-8| > 1$, there is a vertical expansion by a factor of 8 and a reflection across the x-axis due to the negative sign.- **Phase Shift**: - The term $\frac{2\pi}{3}$ inside the cosine function indicates a horizontal phase shift. However, this does not affect vertical expansion or compression.- **Vertical Shift**: - The "+4" outside the cosine function indicates a vertical translation up by 4 units.Overall, the function experiences a vertical expansion due to the amplitude being greater than 1, with an additional reflection and vertical shift.

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Description: Question is attached The image contains a mathematical expression and a question related to it. Here is the transcription:Expression:\[ Y = -8 \cos \left(6x + \frac{2\pi}{3}\right) + 4 \]Question:"Is there a vertical expansion or compression? Pick on

Consider the given function y=-8cos6x+2π3+4. Note that, the general form of the cosine function is…

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The image contains a mathematical expression and a question related to it. Here is the transcription:

Expression:
\[ Y = -8 \cos \left(6x + \frac{2\pi}{3}\right) + 4 \]

Question:
"Is there a vertical expansion or compression? Pick one and explain why."

---

### Explanation

The expression given is a transformation of the cosine function.

- **Vertical Expansion/Compression**:
- The coefficient of the cosine function, $-8$, indicates a vertical transformation. Since $|-8| > 1$, there is a vertical expansion by a factor of 8 and a reflection across the x-axis due to the negative sign.

- **Phase Shift**:
- The term $\frac{2\pi}{3}$ inside the cosine function indicates a horizontal phase shift. However, this does not affect vertical expansion or compression.

- **Vertical Shift**:
- The "+4" outside the cosine function indicates a vertical translation up by 4 units.

Overall, the function experiences a vertical expansion due to the amplitude being greater than 1, with an additional reflection and vertical shift.

Expert Solution

Step 1

Consider the given function $y = - 8 \cos (6 x + \frac{2 π}{3}) + 4$ .

Note that, the general form of the cosine function is $y = A \cos (B x \pm C) \pm D$ .

Where, $|A|$ is amplitude, B - cycles from 0 to $2 π$ . That implies, $period = \frac{2 π}{B}$ .

C - horizontal shift (known as phase shift when B = 1), and D - vertical shift (displacement).

Step 2

Compare the given function $y = - 8 \cos (6 x + \frac{2 π}{3}) + 4$ with $y = A \cos (B x \pm C) \pm D$ .

$A = - 8, B = 6, C = \frac{2 π}{3}, and D = 4$ .

That implies,

$\begin{array}{rcl} Amplitude & = & |A| \\ = & |- 8| \\ = & 8 \end{array}$

$\begin{array}{rcl} Period & = & \frac{2 π}{6} \\ = & \frac{π}{3} \end{array}$

$\begin{array}{rcl} horizontal shift & = & C \\ = & \frac{2 π}{3} \end{array}$

$\begin{array}{rcl} vertical shift & = & D \\ = & 4 \end{array}$

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