To find the vertex of the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can use the formula for the x-coordinate of the vertex, which is given by:
\[ x = -\frac{b}{2a} \]
where \( a = -6 \) and \( b = 3 \).
Calculating the x-coordinate:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we can find the corresponding y-coordinate by substituting \( x = 0.25 \) back into the equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 \] \[ = 0.375 + 8 = 8.375 \]
Thus, the vertex of the parabola is at the point \( (0.25, 8.375) \). Since the coefficient of \( x^2 \) is negative (\( -6 \)), this point is a maximum (high point) of the parabola.
Now to complete the table:
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At \( x = -0.25 \): \[ y = -6(-0.25)^2 + 3(-0.25) + 8 \] \[ = -6(0.0625) - 0.75 + 8 \] \[ = -0.375 - 0.75 + 8 \] \[ = -1.125 + 8 = 6.875 \]
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At \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 \] \[ = -6(0.25) + 1.5 + 8 \] \[ = -1.5 + 1.5 + 8 = 8 \]
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At \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 \] \[ = -6(0.5625) + 2.25 + 8 \] \[ = -3.375 + 2.25 + 8 = 6.875 \]
Now the completed table looks like this:
\[ \begin{array}{|c|c|} \hline x & y \ \hline -1 & -1 \ -0.75 & 2.375 \ -0.5 & 5 \ -0.25 & 6.875 \ 0 & 8 \ 0.25 & 8.375 \ 0.5 & 8 \ 0.75 & 6.875 \ 1 & 5 \ \hline \end{array} \]
Summary of Response Options
Given the calculated points, the correct statement from the options provided is:
(0.25, 8.375) is the high point on the parabola.
None of the other statements provided are correct based on the calculations. Thus, the relevant option is the one regarding \( (0.25, 8.375) \) as the high point on the parabola.
1 answer