To find the high or low point of the parabola described by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can determine the vertex of the parabola. Since the coefficient of \( x^2 \) is negative, the parabola opens downward, indicating that the vertex is a high point.
The x-coordinate of the vertex for a quadratic equation given by \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] In this equation:
- \( a = -6 \)
- \( b = 3 \)
Plugging in those values: \[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we can find the corresponding \( y \)-value by substituting \( x = 0.25 \) back into the equation: \[ y = -6(0.25)^2 + 3(0.25) + 8 \] Calculating \( (0.25)^2 = 0.0625 \): \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 = 8.375 \]
So, the vertex is at \( (0.25, 8.375) \), which is the high point of the parabola.
Now, let's check the given statements:
- \( (0.5, 9) \) is the high point on the parabola. (Incorrect)
- \( (-1, -1) \) is the low point on the parabola. (Incorrect)
- \( (0.25, 8.375) \) is the low point on the parabola. (Incorrect)
- \( (0.25, 8.375) \) is the high point on the parabola. (Correct)
Thus, the correct response is:
(0.25, 8.375) is the high point on the parabola.