To complete the table for the quadratic equation \( y = 3x^2 - 6x + 9 \), we will calculate the value of \( y \) for each value of \( x \).
Let's first fill in the missing \( y \) values from the given \( x \) values in the table.
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When \( x = -3 \): \[ y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \]
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When \( x = -2 \): \[ y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \]
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When \( x = -1 \): \[ y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18 \]
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When \( x = 0 \): \[ y = 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9 \]
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When \( x = 1 \): \[ y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]
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When \( x = 2 \): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \]
Now, let's summarize the completed table:
| x | y | |-----|-------| | -3 | 54 | | -2 | 33 | | -1 | 18 | | 0 | 9 | | 1 | 6 | | 2 | 9 |
Finding the Vertex (High or Low Point)
Since this is a quadratic function of the form \( y = ax^2 + bx + c \) with \( a = 3 > 0 \), it opens upwards. Therefore, it has a low point (vertex).
The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] where \( a = 3 \) and \( b = -6 \): \[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now, we can substitute \( x = 1 \) back into the equation to find the y-coordinate of the vertex: \[ y = 3(1)^2 - 6(1) + 9 = 3 - 6 + 9 = 6 \]
Thus, the low point of the parabola is at:
The low point of this parabola is (1, 6).