To find the high or low point (vertex) of the quadratic equation \( y = -6x^2 + 3x + 8 \), we first need to calculate all the given values and then identify the vertex.
Calculation of Missing 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
Determining the Vertex
For a quadratic function of the form \( ax^2 + bx + c \):
- The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \).
- Here, \( a = -6 \) and \( b = 3 \).
\[ x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25 \]
Finding the y-coordinate of the vertex
Substituting \( x = 0.25 \) into the original equation to find \( y \):
\[ y = -6(0.25)^2 + 3(0.25) + 8 = -6(0.0625) + 0.75 + 8 = -0.375 + 0.75 + 8 = 8.375 \]
Conclusion
The vertex, \( (0.25, 8.375) \), is the high point of the parabola because the coefficient of \( x^2 \) is negative (indicating a downward-opening parabola).
From the options provided:
- (0.25,8.375) is the high point on the parabola.
This response is correct:
\[ \text{(0.25,8.375) is the high point on the parabola.} \]