To find the vertex of the quadratic equation \( y = -6x^2 + 3x + 8 \), we will first calculate the x-coordinate of the vertex using the formula:
\[ x = -\frac{b}{2a} \]
In our equation, \( a = -6 \) and \( b = 3 \). Plugging in these values:
\[ x = -\frac{3}{2(-6)} = \frac{3}{12} = 0.25 \]
Next, we substitute \( x = 0.25 \) back into the equation to find the corresponding y-coordinate:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 = 8.375 \]
Thus, the vertex of the parabola is \( (0.25, 8.375) \).
Since the coefficient of \( x^2 \) in your quadratic equation is negative, the parabola opens downward, meaning the vertex represents the high point of the parabola.
Now, let's fill in the missing values in the table by calculating \( y \) for \( x = -0.25 \), \( 0.5 \), and \( 0.75 \):
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For \( x = -0.25 \): \[ y = -6(-0.25)^2 + 3(-0.25) + 8 \] \[ y = -6(0.0625) - 0.75 + 8 \] \[ y = -0.375 - 0.75 + 8 \] \[ y = -1.125 + 8 = 6.875 \]
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For \( x = 0.5 \): \[ y = -6(0.5)^2 + 3(0.5) + 8 \] \[ y = -6(0.25) + 1.5 + 8 \] \[ y = -1.5 + 1.5 + 8 \] \[ y = 0 + 8 = 8 \]
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For \( x = 0.75 \): \[ y = -6(0.75)^2 + 3(0.75) + 8 \] \[ y = -6(0.5625) + 2.25 + 8 \] \[ y = -3.375 + 2.25 + 8 \] \[ y = -1.125 + 8 = 6.875 \]
Now we can summarize the completed values in the table:
| 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 |
Based on this analysis:
- The point \( (0.25, 8.375) \) is the high point of the parabola.
The correct response from the options provided is:
(0.25, 8.375) is the high point on the parabola.