Let's analyze the graph and answer the given questions based on the visible information.

(a) Minimum and Maximum Average Monthly Precipitation:

From the graph, we observe that the precipitation values range between 2 inches and 12 inches.

• The maximum average monthly precipitation appears to be 12 inches.
• The minimum average monthly precipitation appears to be 2 inches.

(b) Amplitude and its Interpretation:

The amplitude is half the difference between the maximum and minimum values of the function.

$\text{Amplitude} = \frac{\text{Max value} - \text{Min value}}{2} = \frac{12 - 2}{2} = 5$

Interpretation: The amplitude represents how much the precipitation fluctuates from the average value over the period. It indicates that the precipitation oscillates by 5 inches from the central average over the 3-year cycle.

(c) Using the cosine model $f(x) = a \cos(b(x - c)) + d$ to model this graph:

• The amplitude $a$ is 5 (as calculated above).
• The midline $d$ represents the vertical shift, which is the average of the maximum and minimum values: $d = \frac{12 + 2}{2} = 7$
• The period of the function can be seen from the graph: one full cycle is completed from month 0 to month 12, so the period is 12 months. The period is related to $b$ by the formula: $\text{Period} = \frac{2\pi}{b} \Rightarrow 12 = \frac{2\pi}{b} \Rightarrow b = \frac{2\pi}{12} = \frac{\pi}{6}$
• To find the phase shift $c$, we observe that the graph reaches its maximum at around month 3, so the phase shift $c$ is 3.

Therefore, the cosine function modeling the graph is:

$f(x) = 5 \cos\left(\frac{\pi}{6}(x - 3)\right) + 7$

Would you like further clarification on any part of this solution? Here are some related questions for practice:

1. What is the midline of a trigonometric function, and how is it calculated?
2. How does the value of $b$ in the function affect the period of the cosine wave?
3. How do we determine the phase shift in a trigonometric function from a graph?
4. What is the difference between a sine and cosine function graph, and when would you use one over the other?
5. How would the graph change if the amplitude was doubled?

Tip: Always check the graph's symmetry when identifying phase shifts in periodic functions!