To determine the month in which Phoenix received the lowest amount of precipitation using the given quadratic model \( y = -0.04088x^2 + 0.4485x + 1.862 \), we first need to understand that this equation represents a parabola that opens downwards (due to the negative coefficient on the quadratic term).
Steps to Find the Lowest Precipitation:
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Identify the Vertex of the Parabola: Since the parabola opens downwards, the vertex will represent the maximum value of \( y \) (precipitation). The x-value of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] where \( a = -0.04088 \) and \( b = 0.4485 \).
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Calculating the x-value of the Vertex: \[ x = -\frac{0.4485}{2 \times -0.04088} = \frac{0.4485}{0.08176} \approx 5.48 \]
This value (approximately 5.48) suggests that the month with the highest precipitation occurs sometime in June (where \( x = 6 \) corresponds to June).
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Evaluate Precipitation for Each Month: We will calculate the precipitation for each month of the year using the function for \( x \) from 1 to 12 (representing January through December).
- January (x = 1): \[ y = -0.04088(1)^2 + 0.4485(1) + 1.862 \approx 2.26962 \]
- February (x = 2): \[ y = -0.04088(2)^2 + 0.4485(2) + 1.862 \approx 2.20 \]
- March (x = 3): \[ y = -0.04088(3)^2 + 0.4485(3) + 1.862 \approx 2.10 \]
- April (x = 4): \[ y = -0.04088(4)^2 + 0.4485(4) + 1.862 \approx 2.00 \]
- May (x = 5): \[ y = -0.04088(5)^2 + 0.4485(5) + 1.862 \approx 1.86 \]
- June (x = 6): \[ y = -0.04088(6)^2 + 0.4485(6) + 1.862 \approx 1.76 \]
- July (x = 7): \[ y = -0.04088(7)^2 + 0.4485(7) + 1.862 \approx 1.73 \]
- August (x = 8): \[ y = -0.04088(8)^2 + 0.4485(8) + 1.862 \approx 1.70 \]
- September (x = 9): \[ y = -0.04088(9)^2 + 0.4485(9) + 1.862 \approx 2.59 \text{ (Given)} \]
- October (x = 10): \[ y = -0.04088(10)^2 + 0.4485(10) + 1.862 \approx 2.00 \]
- November (x = 11): \[ y = -0.04088(11)^2 + 0.4485(11) + 1.862 \approx 1.70 \]
- December (x = 12): \[ y = -0.04088(12)^2 + 0.4485(12) + 1.862 \approx 1.50 \]
Table of Precipitation:
| Month | Precipitation (inches) | |------------|-------------------------| | January | 2.27 | | February | 2.20 | | March | 2.10 | | April | 2.00 | | May | 1.86 | | June | 1.76 | | July | 1.73 | | August | 1.70 | | September | 2.59 | | October | 2.00 | | November | 1.70 | | December | 1.50 |
Conclusion:
Based on the calculations from the model, the month in which Phoenix received the lowest amount of precipitation is December with approximately 1.50 inches.