To find the values of the high or low point of the parabola defined by the equation \( y = 3x^2 - 6x + 9 \), we can first calculate \( y \) for the missing \( x \) values in the table.
Given Equation:
\[ y = 3x^2 - 6x + 9 \]
Calculating y-values:
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For \( x = -3 \): \[ y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \] (Confirmed as provided)
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For \( x = -2 \): \[ y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \] (Confirmed as provided)
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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 \( x = 2 \): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \] (Confirmed as provided)
Completed Table:
\[ \begin{array}{|c|c|} \hline x & y \ \hline -3 & 54 \ -2 & 33 \ -1 & 18 \ 0 & 9 \ 1 & 6 \ 2 & 9 \ \hline \end{array} \]
Finding the Vertex (High or Low Point):
For a quadratic equation of the form \( ax^2 + bx + c \), the x-coordinate of the vertex (high or low point) can be calculated using: \[ x = -\frac{b}{2a} \] Here, \( a = 3 \) and \( b = -6 \): \[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now we can find \( y \) when \( x = 1 \): \[ y = 3(1)^2 - 6(1) + 9 = 3 - 6 + 9 = 6 \]
Conclusion:
The high or low point of the parabola is at: \[ (1, 6) \]
Thus, the final answer is: \[ \text{The high or low point of this parabola is } (1, 6). \]
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