To find the high or low point (vertex) of the quadratic equation \( y = -6x^2 + 3x + 8 \), we can use the formula for the x-coordinate of the vertex of a parabola, given by:
\[ x = -\frac{b}{2a} \]
where \( a \) and \( b \) are the coefficients from the general form \( y = ax^2 + bx + c \).
In this case, you have:
- \( a = -6 \)
- \( b = 3 \)
Now, substituting these values into the formula:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we need to find the corresponding \( y \)-value by substituting \( x = 0.25 \) back into the quadratic equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \]
Calculating this step-by-step:
- \( (0.25)^2 = 0.0625 \)
- \( -6 \times 0.0625 = -0.375 \)
- \( 3 \times 0.25 = 0.75 \)
- Summing these: \( -0.375 + 0.75 + 8 = 8.375 \)
Thus, the vertex (point where the parabola reaches its maximum because it opens downwards) is:
\[ (0.25, 8.375) \]
Now, reviewing the 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.
Given that \( (0.25, 8.375) \) is indeed the vertex and we previously established that this is the high point because the parabola opens downwards, the correct statement is:
(0.25, 8.375) is the high point on the parabola.