To find the high or low point of the quadratic equation \(y = 3x^2 - 6x + 9\), we can complete the given table of values for \(y\) and then analyze the vertex of the equation, which represents the high or low point of the parabola.
The vertex of the parabola given by the equation \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]
In the equation \(y = 3x^2 - 6x + 9\):
- \(a = 3\)
- \(b = -6\)
Plugging the values into the vertex formula:
\[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now we can find the corresponding \(y\) value by plugging \(x = 1\) back into the equation:
\[ y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]
Thus, the vertex is at the point \((1, 6)\).
To fully complete the table with values of \(y\):
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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 \(x = 1\): \[ y = 6 \quad \text{(calculated above)} \]
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For \(x = 2\): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \]
Hence, the completed table is as follows:
| \(x\) | \(y\) | |-------|-------| | -3 | 54 | | -2 | 33 | | -1 | 18 | | 0 | 9 | | 1 | 6 | | 2 | 9 |
The vertex (also the low point since the parabola opens upwards) of this quadratic function is at \((1, 6)\).