To determine the month in which Phoenix received the lowest amount of precipitation, we will analyze the given quadratic model for precipitation:
[tex]\[ y = -0.04088x^2 + 0.4485x + 1.862 \][/tex]
To find the month with the minimum precipitation, we follow these steps:
1. Determine the critical points: These are found by taking the derivative of the function [tex]\( y \)[/tex] and setting it to zero. The derivative of [tex]\( y \)[/tex] is:
[tex]\[ y' = -0.08176x + 0.4485 \][/tex]
Set the derivative equal to zero to find the critical points:
[tex]\[ -0.08176x + 0.4485 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{0.4485}{0.08176} \approx 5.486 \][/tex]
2. Evaluate the function at the critical point: Substitute [tex]\( x = 5.486 \)[/tex] into the original equation to find the precipitation amount at this point:
[tex]\[ y = -0.04088(5.486)^2 + 0.4485(5.486) + 1.862 \][/tex]
Using the critical point [tex]\( x \)[/tex], we find:
[tex]\[ y \approx 3.092 \text{ inches} \][/tex]
Thus, the month corresponding to [tex]\( x \approx 5.486 \)[/tex], which is approximately 5.5. Considering x = 5 represents May and x = 6 represents June, it suggests that the lowest precipitation occurs around the end of May or beginning of June.
Therefore, the month in which Phoenix received the lowest amount of precipitation is approximately June (the closest to 5.5).
