To find the values for the high or low point of the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can start by completing the table for the specified x-values.
Let's calculate \( y \) for the missing \( x \) values in the table:
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For \( x = -1 \): \[ y = -6(-1)^2 + 3(-1) + 8 = -6(1) - 3 + 8 = -6 - 3 + 8 = -1 \] So, the entry is \((-1, -1)\).
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For \( x = -0.25 \): \[ y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875 \] So, the entry is \((-0.25, 6.875)\).
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For \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8 \] So, the entry is \((0.5, 8)\).
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For \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875 \] So, the entry is \((0.75, 6.875)\).
The completed table is as follows:
| \( x \) | \( y \) | |-----------|-------------| | -1 | -1 | | -0.75 | 2.375 | | -0.5 | 5 | | -0.25 | 6.875 | | 0 | 8 | | 0.25 | 8.375 | | 0.5 | 8 | | 0.75 | 6.875 | | 1 | 5 |
Finding the High or Low Point
Since the coefficient of \( x^2 \) in the equation \( y = -6x^2 + 3x + 8 \) is negative (-6), this indicates that the parabola opens downwards, and thus, it will have a high point (the vertex).
The vertex of a parabola given by \( ax^2 + bx + c \) is found at: \[ x = -\frac{b}{2a} \]
Here, \( a = -6 \) and \( b = 3 \):
\[ x = -\frac{3}{2 \times -6} = \frac{3}{12} = 0.25 \]
Let’s calculate \( y \) at this \( x \):
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 = 8.375 \]
Thus, the vertex (high point) is at \( (0.25, 8.375) \).
Conclusion
- The point \( (0.25, 8.375) \) is the high point.
- Therefore, the correct response from the options given is:
\((0.25, 8.375)\) is the high point on the parabola.