To find the month in which Phoenix, Arizona, received the lowest amount of precipitation, we need to analyze the given quadratic function model that represents the precipitation in Phoenix:
[tex]\[ y = -0.04088 x^2 + 0.4485 x + 1.862 \][/tex]
This quadratic function is in the standard form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -0.04088 \)[/tex], [tex]\( b = 0.4485 \)[/tex], and [tex]\( c = 1.862 \)[/tex]. Since the leading coefficient [tex]\( a \)[/tex] is negative, the parabola opens downward, meaning it has a maximum point at the vertex.
To find the vertex of the quadratic function, we use the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ x = -\frac{0.4485}{2(-0.04088)} \][/tex]
[tex]\[ x = -\frac{0.4485}{-0.08176} \][/tex]
[tex]\[ x = \frac{0.4485}{0.08176} \][/tex]
[tex]\[ x \approx 5.486 \][/tex]
Since [tex]\( x \)[/tex] represents the month, and we usually consider months as whole numbers, we round [tex]\( x \)[/tex] to the nearest integer. Thus, [tex]\( x \approx 5 \)[/tex].
So, the month with the lowest precipitation is [tex]\( x = 5 \)[/tex], which corresponds to May.
Next, we determine the amount of precipitation in May by substituting [tex]\( x = 5 \)[/tex] back into the original quadratic equation:
[tex]\[ y = -0.04088 (5)^2 + 0.4485 (5) + 1.862 \][/tex]
[tex]\[ y = -0.04088 (25) + 0.4485 (5) + 1.862 \][/tex]
[tex]\[ y = -1.022 + 2.2425 + 1.862 \][/tex]
[tex]\[ y \approx 3.0825 \][/tex]
Therefore, the lowest precipitation in Phoenix occurred in May, with approximately 3.0825 inches.
Here is the updated table with the precipitation for May filled in:
\begin{tabular}{|l|l|}
\hline
Month ( [tex]\(x\)[/tex] ) & Precipitation \\
\hline
January & 2.27 inches \\
\hline
February & ? \\
\hline
March & ? \\
\hline
April & ? \\
\hline
May & 3.0825 inches \\
\hline
June & ? \\
\hline
July & ? \\
\hline
August & ? \\
\hline
September & 2.59 inches \\
\hline
October & ? \\
\hline
November & ? \\
\hline
December & ? \\
\hline
\end{tabular}
So, to answer the question: Phoenix received the lowest amount of precipitation in the month of May.
