To determine the high or low point of the parabola given by the equation \( y = -6x^2 + 3x + 8 \), we need to find the vertex of the parabola. The vertex formula for a quadratic equation in standard form \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
Here, \( a = -6 \) and \( b = 3 \). Plugging in these values:
\[ x = -\frac{3}{2 \cdot -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we find the corresponding \( y \) value by substituting \( x = 0.25 \) back into the equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 8.375 \]
Thus, the vertex of the parabola, which represents the maximum (since the parabola opens downwards) or minimum point, is at \( (0.25, 8.375) \).
Since the parabola opens downwards (the coefficient of \( x^2 \) is negative), \( (0.25, 8.375) \) is the high point on the parabola.
Now, we can evaluate the options provided:
- (0.25, 8.375) is the high point on the parabola: This is correct.
- (0.25, 8.375) is the low point on the parabola: This is incorrect.
- (0.5, 9) is the high point on the parabola: This is incorrect.
- (−1, −1) is the low point on the parabola: This is incorrect.
Therefore, the correct response is:
(0.25, 8.375) is the high point on the parabola.
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