To determine whether a point is a high or low point on the parabola represented by the quadratic equation \( y = -6x^2 + 3x + 8 \), we need to find the vertex of the parabola. The vertex gives us the maximum or minimum point, depending on the direction the parabola opens.
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Finding the Vertex: For a quadratic equation in the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = -6 \) and \( b = 3 \).
Substituting these values into the formula: \[ x = -\frac{3}{2 \times -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
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Calculating the y-coordinate at \( x = 0.25 \): Now we can find the y-coordinate by substituting \( x = 0.25 \) back into the original equation: \[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ = -6(0.0625) + 0.75 + 8 \] \[ = -0.375 + 0.75 + 8 \] \[ = 0.375 + 8 = 8.375 \]
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Vertex of the Parabola: The vertex of the parabola is at \( (0.25, 8.375) \). Since the coefficient of \( x^2 \) (which is \(-6\)) is negative, this means the parabola opens downward; hence, the vertex represents the high point of the parabola.
Conclusion:
Looking at the options provided:
- \( (0.25, 8.375) \) is the high point of the parabola.
Correct Response: \( (0.25, 8.375) \) is the high point on the parabola.