To complete the table using the quadratic equation \( y = 3x^2 - 6x + 9 \), we need to calculate the values of \( y \) for each \( x \) value provided.
Let's compute \( y \) for \( x = -3, -2, -1, 0, 1 \), and \( 2 \):
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For \( x = -3 \): \[ y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \]
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For \( x = -2 \): \[ y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \]
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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 \( x = 2 \): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \]
Now, we can complete the table:
\[ \begin{array}{|c|c|} \hline x & y \ \hline -3 & 54 \ -2 & 33 \ -1 & 18 \ 0 & 9 \ 1 & 6 \ 2 & 9 \ \hline \end{array} \]
Next, to find the high or low point of the parabola, we should identify the vertex of the parabola given by the formula for the vertex of a quadratic \( y = ax^2 + bx + c \), which occurs at \( x = -\frac{b}{2a} \).
For the equation \( y = 3x^2 - 6x + 9 \):
- \( a = 3 \)
- \( b = -6 \)
Calculating the x-coordinate of the vertex:
\[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now substituting \( x = 1 \) back into the equation to find the y-coordinate:
\[ y = 3(1)^2 - 6(1) + 9 = 3 - 6 + 9 = 6 \]
Thus, the high or low point of the parabola (the vertex) is:
\[ \text{The high or low point of this parabola is } (1, 6). \]
Since the coefficient of \( x^2 \) (which is 3) is positive, the parabola opens upwards, meaning this point is a low point.