To determine the high or low point (the vertex) of the parabola given by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can use the vertex formula. The x-coordinate of the vertex for a quadratic equation of the form \( y = ax^2 + bx + c \) is found using:
\[ x = -\frac{b}{2a} \]
In our equation, \( a = -6 \), \( b = 3 \), and \( c = 8 \).
Let's calculate the x-coordinate of the vertex:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Next, we need to calculate the y-coordinate of the vertex by substituting \( x = 0.25 \) back into the original equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \]
Calculating \( (0.25)^2 \):
\[ (0.25)^2 = 0.0625 \]
Substituting that into the equation:
\[ y = -6(0.0625) + 3(0.25) + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 = 8.375 \]
So the vertex, which is the high point of the parabola (because the parabola opens downwards, indicated by a negative \( a \)), is at:
\[ (0.25, 8.375) \]
Therefore, let's evaluate the options given:
A: (0.5, 9) is the high point on the parabola. - Incorrect.
B: (0.25, 8.375) is the low point on the parabola. - Incorrect, it's a high point.
C: (-1, -1) is the low point on the parabola. - Incorrect.
D: (0.25, 8.375) is the high point on the parabola. - Correct.
The correct option is D: (0.25, 8.375) is the high point on the parabola.