To analyze the given quadratic function representing U.S. crude oil production, we need to perform several calculations. The function is given by:
\[ y = -0.002x^2 + 0.033x + 1.957 \]
where \( x \) is the number of years after 2015.
a. Find and interpret the vertex of the graph of this model.
The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In our case, \( a = -0.002 \) and \( b = 0.033 \).
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Find \( x \): \[ x = -\frac{0.033}{2 \times -0.002} = -\frac{0.033}{-0.004} = 8.25 \]
This means that the vertex occurs at \( x = 8.25 \), which corresponds to the year \( 2015 + 8.25 = 2023.25 \) (approximately mid-2023).
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Find \( y \) at the vertex: Now, we substitute \( x = 8.25 \) back into the original equation to find \( y \): \[ y = -0.002(8.25)^2 + 0.033(8.25) + 1.957 \] \[ y = -0.002(68.0625) + 0.27225 + 1.957 \] \[ y = -0.136125 + 0.27225 + 1.957 \] \[ y \approx 2.093875 \text{ billion barrels} \]
Interpretation: The vertex of the function is approximately \((8.25, 2.093875)\). This means that U.S. crude oil production is projected to reach its maximum value of approximately 2.094 billion barrels around mid-2023.
b. What does the model predict the crude oil production will be in 2038?
To find the projection for 2038, we need to determine \( x \) as follows: \[ x = 2038 - 2015 = 23 \]
Now we evaluate \( y \) at \( x = 23 \): \[ y = -0.002(23)^2 + 0.033(23) + 1.957 \] \[ y = -0.002(529) + 0.759 + 1.957 \] \[ y = -1.058 + 0.759 + 1.957 \] \[ y \approx 1.658 \text{ billion barrels} \]
Prediction: The model predicts that crude oil production in 2038 will be approximately 1.658 billion barrels.
c. Graph the function for the years 2015 to 2035.
To graph the function from 2015 to 2035, we can calculate the \( y \) values for various \( x \) values in this range:
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When \( x = 0 \) (2015): \[ y = 1.957 \text{ billion barrels} \]
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When \( x = 5 \) (2020): \[ y = -0.002(5)^2 + 0.033(5) + 1.957 = -0.05 + 0.165 + 1.957 = 2.072 \text{ billion barrels} \]
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When \( x = 10 \) (2025): \[ y = -0.002(10)^2 + 0.033(10) + 1.957 = -0.2 + 0.33 + 1.957 = 2.087 \text{ billion barrels} \]
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When \( x = 15 \) (2030): \[ y = -0.002(15)^2 + 0.033(15) + 1.957 = -0.45 + 0.495 + 1.957 = 2.002 \text{ billion barrels} \]
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When \( x = 20 \) (2035): \[ y = -0.002(20)^2 + 0.033(20) + 1.957 = -0.8 + 0.66 + 1.957 = 1.817 \text{ billion barrels} \]
The points you would plot based on the calculations above are approximately:
- \( (0, 1.957) \)
- \( (5, 2.072) \)
- \( (10, 2.087) \)
- \( (15, 2.002) \)
- \( (20, 1.817) \)
Graph: The graph will be a downward-opening parabola with the vertex at approximately (8.25, 2.094).
To visualize, you would plot these points on a graph with the x-axis covering the range from 0 (2015) to 20 (2035) and the y-axis showing the production levels in billions of barrels.
Summary
- The vertex is located at about 2023.25, predicting maximum production around 2.094 billion barrels.
- The projection for 2038 is about 1.658 billion barrels.
- The values calculated provide a basis for graphing the function from 2015 to 2035.
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