To write the equation of a cosine function with the given characteristics:

• Amplitude: $3$
• Midline: $y = 4$
• Period: $2\pi$

The general form of a cosine function is:

$y = A \cos(Bx + C) + D$

Where:

• $A$ is the amplitude.
• $B$ is related to the period by the formula $\text{Period} = \frac{2\pi}{B}$.
• $D$ is the vertical shift (midline).

Given the information:

• $A = 3$
• The period is $2\pi$, so $B = 1$ (since $\frac{2\pi}{B} = 2\pi$ implies $B = 1$).
• $D = 4$

Assuming there is no phase shift (i.e., $C = 0$), the equation of the cosine function is:

$y = 3 \cos(x) + 4$

Would you like to dive deeper into any of these details or have further questions?

Here are 5 related questions to consider:

1. How would the function change if the amplitude were reduced to 1?
2. What is the effect on the graph if the midline is changed to $y = 0$?
3. How would the equation look if the period were $\pi$ instead of $2\pi$?
4. What would the phase shift be if $C = \frac{\pi}{2}$?
5. How do you determine the maximum and minimum points of this cosine function?

Tip: Remember, for trigonometric functions, changing the period affects the frequency of the function, while changing the midline affects its vertical position on the graph.