To solve for the missing values in the table and find the vertex (high or low point) of the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we can start by completing the table for the values of \( x \).
- Calculate the missing \( y \) values:
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 \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 = -6(0.5625) + 2.25 + 8 = -3.375 + 2.25 + 8 = 6.875 \]
The completed table is as follows:
| \( x \) | \( y \) | |----------|-------------| | -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 |
- Finding the Vertex (High or Low Point):
The vertex of the parabola in the standard form \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] For the equation \( y = -6x^2 + 3x + 8 \),
- \( a = -6 \)
- \( b = 3 \)
Calculating the vertex \( x \): \[ x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25 \]
Now, substituting \( x = 0.25 \) back into the equation to find the \( y \)-value: \[ y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375 \]
Thus, the vertex is at \( (0.25, 8.375) \).
Since the coefficient \( a = -6 \) is negative, the parabola opens downwards, indicating that \( (0.25, 8.375) \) is the high point on the parabola.
- Conclusion:
Among the options provided:
- (0.25, 8.375) is the high point on the parabola. This is the correct choice.
So, the answer is:
- (0.25, 8.375) is the high point on the parabola.
1 answer