Let's break down each trigonometric function to find the amplitude, period, and phase shift, then summarize for each case.

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a) $y = 3 \cos \left( x + \frac{\pi}{4} \right)$

Step 1: Amplitude
The amplitude is the coefficient of the cosine function.
$\text{Amplitude} = 3$

Step 2: Period
The standard period of $\cos(x)$ is $2\pi$. Since there's no coefficient of $x$, the period remains the same:
$\text{Period} = 2\pi$

Step 3: Phase Shift
To find the phase shift, we look at the expression inside the cosine. The general form is $\cos(bx + c)$. The phase shift is calculated as:
$\text{Phase Shift} = -\frac{c}{b}$ Here, $c = \frac{\pi}{4}$ and $b = 1$, so:
$\text{Phase Shift} = -\frac{\frac{\pi}{4}}{1} = -\frac{\pi}{4}$

So, this function has a phase shift of $-\frac{\pi}{4}$, meaning it's shifted to the left by $\frac{\pi}{4}$.

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b) $y = \sin \left( \frac{1}{2} \left( x - \frac{\pi}{4} \right) \right)$

Step 1: Amplitude
The amplitude is 1 since the coefficient of the sine function is 1.

Step 2: Period
The period of the sine function is $2\pi$, but here the function has a coefficient of $\frac{1}{2}$ inside the parentheses. The period is calculated as:
$\text{Period} = \frac{2\pi}{b}$ where $b = \frac{1}{2}$, so:
$\text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi$

Step 3: Phase Shift
The phase shift is given by $\frac{c}{b}$, where $c = \frac{\pi}{4}$ and $b = \frac{1}{2}$:
$\text{Phase Shift} = \frac{\frac{\pi}{4}}{\frac{1}{2}} = \frac{\pi}{2}$

So, the phase shift is $\frac{\pi}{2}$ to the right.

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c) $y = 2 \sin(2x - \pi)$

Step 1: Amplitude
The amplitude is 2 since the coefficient of the sine function is 2.
$\text{Amplitude} = 2$

Step 2: Period
The period is calculated as:
$\text{Period} = \frac{2\pi}{b}$ Here, $b = 2$, so:
$\text{Period} = \frac{2\pi}{2} = \pi$

Step 3: Phase Shift
The phase shift is given by $\frac{c}{b}$, where $c = \pi$ and $b = 2$:
$\text{Phase Shift} = \frac{\pi}{2}$

So, the phase shift is $\frac{\pi}{2}$ to the right.

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Summary:

1. a) $y = 3 \cos \left( x + \frac{\pi}{4} \right)$

   • Amplitude: $3$
   • Period: $2\pi$
   • Phase Shift: $-\frac{\pi}{4}$ (left)

2. b) $y = \sin \left( \frac{1}{2} \left( x - \frac{\pi}{4} \right) \right)$

   • Amplitude: $1$
   • Period: $4\pi$
   • Phase Shift: $\frac{\pi}{2}$ (right)

3. c) $y = 2 \sin(2x - \pi)$

   • Amplitude: $2$
   • Period: $\pi$
   • Phase Shift: $\frac{\pi}{2}$ (right)

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Would you like me to provide graphs for each function or additional details on these results?

Here are some follow-up questions to deepen your understanding:

1. How does changing the amplitude affect the shape of the graph?
2. What happens to the period if the coefficient of $x$ inside the sine or cosine function increases?
3. How does the phase shift impact the starting point of the graph?
4. Can you explain how the period differs between sine and cosine functions?
5. How would you graph a cosine function with a negative amplitude?

Tip: The period of a trigonometric function depends on the frequency, which is the coefficient of $x$ inside the function.