To analyze the transformations that turn the graph of $y = \cos(x)$ into the graph of $y = 6 + 3 \cos\left(x + \frac{\pi}{4}\right)$, let's break down the changes:

1. Horizontal Shift: The expression inside the cosine, $x + \frac{\pi}{4}$, represents a horizontal shift. Since the $+\frac{\pi}{4}$ is added to $x$, this indicates a shift to the left by $\frac{\pi}{4}$ units.
   So, the graph is shifted to the left by $\frac{\pi}{4}$ units.

2. Vertical Stretch and Reflection: The coefficient $3$ in front of the cosine function represents a vertical stretch by a factor of 3. The graph is stretched by 3 times the usual amplitude of the cosine function.
   There is no reflection because the coefficient is positive.

3. Vertical Shift: The constant $6$ outside the cosine function represents a vertical shift. Since it is added to the entire function, it shifts the graph up by 6 units.
   So, the graph is shifted up by 6 units.

Summary:

• The graph of $y = 6 + 3 \cos\left(x + \frac{\pi}{4}\right)$ is obtained by:
  1. Shifting the graph of $y = \cos(x)$ left by $\frac{\pi}{4}$ units.
  2. Stretching it vertically by a factor of 3.
  3. Shifting it up by 6 units.

Let me know if you'd like further details or have any questions!

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Here are some related questions:

1. What is the effect of a negative sign in front of the cosine function?
2. How does the period of the cosine function change if the equation is $y = 3 \cos(2x)$?
3. What happens if you shift the graph of $y = \cos(x)$ by a fraction of $\pi$ units?
4. How does the phase shift affect the graph of the sine function?
5. What is the general form of a sinusoidal function, and how do you interpret each parameter?

Tip: When analyzing transformations of sinusoidal functions like cosine or sine, remember that:

• A positive inside shift (e.g., $x + c$) shifts the graph to the left.
• A negative inside shift (e.g., $x - c$) shifts the graph to the right.