To determine the month in which Phoenix received the lowest amount of precipitation, we need to find the minimum point of the quadratic function modeling the precipitation. The given function is:
[tex]\[ y = -0.04088x^2 + 0.44x + 1.862 \][/tex]
This is a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -0.04088 \)[/tex]
- [tex]\( b = 0.44 \)[/tex]
- [tex]\( c = 1.862 \)[/tex]
The quadratic function opens downward because the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a \)[/tex]) is negative. The minimum point of a quadratic function that opens downward is found at its vertex. The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{0.44}{2(-0.04088)} \][/tex]
[tex]\[
x = \frac{0.44}{0.08176}
\][/tex]
[tex]\[
x = 5.38160469667319
\][/tex]
This value represents the month in which Phoenix received the lowest amount of precipitation. Since January is represented by [tex]\( x = 1 \)[/tex], February by [tex]\( x = 2 \)[/tex], and so on, we can interpret [tex]\( x = 5.38160469667319 \)[/tex] as occurring between May ( [tex]\( x = 5 \)[/tex] ) and June ( [tex]\( x = 6 \)[/tex] ), slightly past May.
Summarizing, the lowest precipitation in Phoenix occurred just after May. Based on this result, we infer that the minimum precipitation occurred around the end of May or the beginning of June.
