To complete the table for the quadratic equation \( y = 3x^2 - 6x + 9 \), we can substitute the given \( x \) values into the equation to find the corresponding \( y \) values.
Given Values
- The equation is: \( y = 3x^2 - 6x + 9 \).
- We have \( x \) values of -3, -2, -1, 0, 1, and 2, and we need to find the corresponding \( y \) values.
Calculating the Missing \( y \) Values
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For \( x = -3 \): \[ y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \] (This value is already provided in the table.)
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For \( x = -2 \): \[ y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \] (This value is already provided in the table.)
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For \( x = -1 \): \[ y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18 \]
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For \( x = 0 \): \[ y = 3(0)^2 - 6(0) + 9 = 0 - 0 + 9 = 9 \]
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For \( x = 1 \): \[ y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]
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For \( x = 2 \): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \] (This value is already provided in the table.)
Completed Table of Values
Now we can summarize the completed table:
| x | y | |-----|-----| | -3 | 54 | | -2 | 33 | | -1 | 18 | | 0 | 9 | | 1 | 6 | | 2 | 9 |
Finding the Vertex (High/Low Point of the Parabola)
The vertex of a quadratic equation in the form \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).
In this case:
- \( a = 3 \)
- \( b = -6 \)
Calculating the vertex \( x \): \[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now, substituting \( x = 1 \) back into the equation to find the \( y \) value at the vertex: \[ y = 3(1)^2 - 6(1) + 9 = 3 - 6 + 9 = 6 \]
Conclusion
The high or low point of the parabola is at \( (1, 6) \). Thus, the vertex, which represents the minimum point in this case, is at \( (1, 6) \).