To determine the vertex of the parabola given by the equation \(y = -6x^2 + 3x + 8\), we need to find its maximum or minimum point. Since the coefficient of \(x^2\) is negative, this parabola opens downwards, meaning it has a maximum point (high point).
The x-coordinate of the vertex (high point) of a quadratic equation in the form \(y = ax^2 + bx + c\) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
Where \(a = -6\) and \(b = 3\):
\[ x = -\frac{3}{2 \times -6} = -\frac{3}{-12} = \frac{1}{4} = 0.25 \]
Now, we can find the corresponding y-value by substituting \(x = 0.25\) back into the equation:
\[ y = -6(0.25)^2 + 3(0.25) + 8 \] \[ y = -6(0.0625) + 0.75 + 8 \] \[ y = -0.375 + 0.75 + 8 \] \[ y = 0.375 + 8 = 8.375 \]
Thus, the vertex of the parabola is at the point \((0.25, 8.375)\), which is indeed the high point on the parabola.
Now, let's fill in the missing values in the table.
For \(x = -0.25\):
\[ y = -6(-0.25)^2 + 3(-0.25) + 8 \] \[ y = -6(0.0625) - 0.75 + 8 \] \[ y = -0.375 - 0.75 + 8 = -1.125 + 8 = 6.875 \]
For \(x = 0.5\):
\[ y = -6(0.5)^2 + 3(0.5) + 8 \] \[ y = -6(0.25) + 1.5 + 8 \] \[ y = -1.5 + 1.5 + 8 = 8 \]
For \(x = 0.75\):
\[ y = -6(0.75)^2 + 3(0.75) + 8 \] \[ y = -6(0.5625) + 2.25 + 8 \] \[ y = -3.375 + 2.25 + 8 = -1.125 + 8 = 6.625 \]
Now, the completed table looks like this:
\[ \begin{array}{|c|c|} \hline x & y \ \hline -1 & -1 \ -0.75 & 2.375 \ -0.5 & 5 \ -0.25 & 6.875 \ 0 & 8 \ 0.25 & 8.375 \ 0.5 & 8 \ 0.75 & 6.625 \ 1 & 5 \ \hline \end{array} \]
Now, reviewing the options given:
- \( (0.25, 8.375) \) is the high point on the parabola. (Correct)
- \( (−1,−1) \) is the low point on the parabola. (Incorrect)
- \( (0.5, 9) \) is the high point on the parabola. (Incorrect)
- \( (0.25, 8.375) \) is the low point on the parabola. (Incorrect)
The only correct statement is that \( (0.25, 8.375) \) is the high point on the parabola.
1 answer