To complete the table and find the high or low point of the quadratic equation \( y = -6x^2 + 3x + 8 \), we can calculate the missing values step-by-step.
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Calculate \( y \) for \( x = -0.25 \): \[ y = -6(-0.25)^2 + 3(-0.25) + 8 \] \[ = -6(0.0625) - 0.75 + 8 \] \[ = -0.375 - 0.75 + 8 \] \[ = 6.875 \]
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Calculate \( y \) for \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 \] \[ = -6(0.25) + 1.5 + 8 \] \[ = -1.5 + 1.5 + 8 \] \[ = 8 \]
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Calculate \( y \) for \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 \] \[ = -6(0.5625) + 2.25 + 8 \] \[ = -3.375 + 2.25 + 8 \] \[ = 6.875 \]
Now, the completed table for the values of \( y \) at those respective \( x \) values is:
\[ \begin{array}{|c|c|} \hline x & y \ \hline -1 & -1 \ -0.75 & 2.375 \ -0.5 & 5 \ -0.25 & 6.875 \ 0 & 8 \ 0.25 & 8.375 \ 0.5 & 8 \ 0.75 & 6.875 \ 1 & 5 \ \hline \end{array} \]
To find the high or low point and interpret the results:
- The parabola opens downward since the coefficient of \( x^2 \) is negative (\( -6 \)).
- The vertex of a downward-opening parabola represents the highest point.
To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \) where \( a = -6 \) and \( b = 3 \): \[ x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25 \]
Now, substituting \( x = 0.25 \) back into the quadratic equation to find the maximum \( y \): \[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 = 8.375 \]
So, the vertex (high point) of the parabola is \( (0.25, 8.375) \).
In conclusion, the correct response is:
(0.25, 8.375) is the high point on the parabola.
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