To find the high or low point (the vertex) of the parabola given by the equation \( y = -6x^2 + 3x + 8 \), we can use the formula for the x-coordinate of the vertex of a quadratic equation in the standard form \( y = ax^2 + bx + c \):
\[ x = -\frac{b}{2a} \]
Here, \( a = -6 \) and \( b = 3 \).
Calculating the x-coordinate of the vertex:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we can substitute \( x = 0.25 \) into the equation to find the y-coordinate of the vertex:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 \] \[ = 0.375 + 8 \] \[ = 8.375 \]
So the vertex of the parabola, which is the lowest point because the coefficient of \( x^2 \) is negative (indicating the parabola opens downward), is \( (0.25, 8.375) \).
Now, let's complete the missing values in the table:
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When \( x = -1 \): \[ y = -6(-1)^2 + 3(-1) + 8 = -6(1) - 3 + 8 = -6 - 3 + 8 = -1 \]
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When \( x = -0.25 \): \[ y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875 \]
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When \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8 \]
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When \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875 \]
With these calculations, the completed table looks 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 |
Considering the information given, the correct response is:
(0.25, 8.375) is the low point on the parabola.
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