To find the vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \), we can use the formula for the vertex of a parabola given by the equation \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 4 \) and \( b = 3 \). Plugging in these values:
\[ x = -\frac{3}{2(4)} = -\frac{3}{8} \]
Now, we can find the y-coordinate of the vertex by substituting \( x = -\frac{3}{8} \) back into the equation:
\[ y = 4\left(-\frac{3}{8}\right)^2 + 3\left(-\frac{3}{8}\right) - 8 \]
Calculating \( \left(-\frac{3}{8}\right)^2 = \frac{9}{64} \):
\[ y = 4\left(\frac{9}{64}\right) + 3\left(-\frac{3}{8}\right) - 8 \] \[ y = \frac{36}{64} - \frac{9}{8} - 8 \] \[ y = \frac{36}{64} - \frac{72}{64} - \frac{512}{64} \] \[ y = \frac{36 - 72 - 512}{64} = \frac{-548}{64} = -\frac{137}{16} \approx -8.5625 \]
So the vertex of the quadratic function is:
\[ \left(-\frac{3}{8}, -\frac{137}{16}\right) \]
Among the provided options, the closest value to the calculated vertex \( \left(-0.375, -8.563\right) \) is:
(−0.375,−8.563).