To determine the high or low point of the parabola defined by the equation \( y = -6x^2 + 3x + 8 \), we need to find the vertex of the parabola. The vertex will give us the maximum or minimum value of the quadratic function since it opens downwards (the coefficient of \( x^2 \) is negative).
The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
where \( a = -6 \) and \( b = 3 \). Substituting these values into the formula:
\[ x = -\frac{3}{2(-6)} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we can find the corresponding y-coordinate by substituting \( x = 0.25 \) back into the quadratic equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \]
Calculating \( (0.25)^2 \):
\[ (0.25)^2 = 0.0625 \]
Thus, we have:
\[ y = -6(0.0625) + 3(0.25) + 8 = -0.375 + 0.75 + 8 = 8.375 \]
So the vertex of the parabola is at the point \( (0.25, 8.375) \).
Since the parabola opens downwards (as indicated by the negative \( a \)), this means that \( (0.25, 8.375) \) is the maximum point (or high point) of the parabola.
Now, let's fill in the missing values in the table:
- For \( x = -0.25 \):
\[ y = -6(-0.25)^2 + 3(-0.25) + 8 = -6(0.0625) - 0.75 + 8 = -0.375 - 0.75 + 8 = 6.875 \]
- For \( x = 0.5 \):
\[ y = -6(0.5)^2 + 3(0.5) + 8 = -6(0.25) + 1.5 + 8 = -1.5 + 1.5 + 8 = 8 \]
- For \( x = 0.75 \):
\[ y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875 \]
Now we have our completed table of values:
| x | y | |------|-------| | -1 | -1 | | -0.75| 2.375 | | -0.5 | 5 | | -0.25| 6.875 | | 0 | 8 | | 0.25 | 8.375 | | 0.5 | 8 | | 0.75 | 6.875 | | 1 | 5 |
Now, based on this information, the correct response is:
(0.25, 8.375) is the high point on the parabola.
Thus the correct response is:
(0.25, 8.375) is the high point on the parabola.