The problem given involves analyzing the graph of a cosine function and determining various parameters, such as amplitude, period, phase shift, and angular frequency, based on the function:

$y = -2 \cos(kx - \alpha) + C$

The graph has some specific information, which includes points like $60^\circ$, $210^\circ$, and other key data, and calculations have been started on the right-hand side of the image.

Here’s what we can deduce and verify from the information:

1. Amplitude (A): From the graph and the function form, the amplitude is the absolute value of the coefficient of the cosine function, which is 2. This matches the value mentioned $A = 2$.

2. Vertical Shift (C): The graph shows that the midline seems to be around $y = 3$. Therefore, $C = 3$.

3. Period ($\lambda$): The graph passes through a full cycle starting at $60^\circ$ and ending at $660^\circ$, so the period is $\lambda = 600^\circ$ (as mentioned in the solution).

4. Angular frequency (k): The angular frequency $k$ is related to the period by the formula:

   $k = \frac{360^\circ}{\text{Period}}$ Substituting the period $600^\circ$:

   $k = \frac{360^\circ}{600^\circ} = \frac{3}{5}$

5. Phase shift ($\alpha$): The phase shift can be deduced from the shift of the graph from the origin. It appears to be $\alpha = 60^\circ$, but you can use the relationship $\alpha = \frac{9}{5}$.

Summary of the parameters:

• Amplitude $A = 2$
• Vertical Shift $C = 3$
• Period $\lambda = 600^\circ$
• Angular frequency $k = \frac{3}{5}$
• Phase shift $\alpha = \frac{9}{5}$

Do you need further details on any part of this solution or more explanation?

Here are 5 related questions for practice:

1. What would the graph look like if the amplitude were changed to 4?
2. How would the graph change if the period were doubled?
3. Can you express the function in radians instead of degrees?
4. What is the phase shift in terms of radians?
5. What happens to the graph if $C = -3$ instead of $C = 3$?

Tip:

When analyzing cosine or sine graphs, always look at the amplitude, period, phase shift, and vertical shift as the four key parameters defining the shape of the wave.