The given function is:

$y = -\cos\left[\frac{1}{2}(x - 7)\right]$

Let's break this down:

1. Amplitude: The amplitude of the cosine function, represented by the coefficient in front of $\cos$, is 1. However, since there's a negative sign, the function reflects across the x-axis. Therefore, the graph will have an amplitude of 1, but it will be inverted.

2. Period: The period of a basic cosine function is $2\pi$. When the function is scaled by $\frac{1}{2}$ inside the cosine, the period changes. The new period is calculated as:

$\text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi$

3. Horizontal Shift: The horizontal shift is determined by the $x - 7$ inside the cosine function. The function is shifted to the right by 7 units.

4. Vertical Shift: There is no vertical shift, as there is no constant added outside the cosine function.

So, the function describes a cosine curve with:

• An amplitude of 1, but inverted.
• A period of $4\pi$.
• A horizontal shift of 7 units to the right.

Would you like a graph or further details on how these transformations affect the graph?

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Here are some follow-up questions based on this function:

1. What would the function look like if the coefficient of $\cos$ was $2$?
2. How does the graph change if the period is $\pi$?
3. What effect does a vertical shift of 3 units upwards have on the function?
4. Can you explain the difference between the sine and cosine function graphically?
5. How would the function look if the argument inside the cosine was squared?

Tip: Always check the signs of coefficients inside and outside the function to understand reflections and shifts!