To determine the high or low point of the parabola represented by the quadratic equation \(y = -6x^2 + 3x + 8\), we first need to find the vertex of the parabola. Since the coefficient of \(x^2\) is negative, the parabola opens downward, which means it will have a high point (the vertex).
The formula to find the x-coordinate of the vertex of a quadratic equation in the form \(y = ax^2 + bx + c\) is given by:
\[ x = -\frac{b}{2a} \]
Here, \(a = -6\) and \(b = 3\). Plugging in these values:
\[ x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25 \]
Now, to find the corresponding y-coordinate, we substitute \(x = 0.25\) back into the equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 8.375 \]
The vertex, and therefore the high point, is at the coordinates \((0.25, 8.375)\).
Now, checking the provided options:
- (0.25, 8.375) is the high point on the parabola. - This is correct.
- (−1, −1) is the low point on the parabola. - Incorrect, as this point is not the lowest point.
- (0.5, 9) is the high point on the parabola. - Incorrect.
- (0.25, 8.375) is the low point on the parabola. - Incorrect.
So the correct response is:
(0.25, 8.375) is the high point on the parabola.