To determine the high or low point (vertex) of the quadratic equation \( y = -6x^2 + 3x + 8 \), we first need to confirm that this equation represents a parabola that opens downward (since the coefficient of \( x^2 \) is negative).
The vertex of a parabola given in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
For the quadratic equation \( y = -6x^2 + 3x + 8 \):
- \( a = -6 \)
- \( b = 3 \)
Plugging in these values:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now we substitute \( x = 0.25 \) back into the equation to find the corresponding \( y \) value:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 \] \[ y = 8.375 \]
So the vertex (or the high point of the parabola) is at \( (0.25, 8.375) \).
Now let's review the provided options:
- \((-1, -1)\) is the low point on the parabola.
- \((0.5, 9)\) is the high point on the parabola.
- \((0.25, 8.375)\) is the low point on the parabola.
- \((0.25, 8.375)\) is the high point on the parabola.
Since we found that the vertex is at \( (0.25, 8.375) \) and the parabola opens downward, this means that \( (0.25, 8.375) \) is the high point, not the low point.
Therefore, the correct response is:
\((0.25, 8.375)\) is the high point on the parabola.