To find the vertex of the quadratic equation \(y = 4x^2 + 3x - 8\), we can use the formula for the vertex of a quadratic function \(y = ax^2 + bx + c\). The x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
In this quadratic, \(a = 4\) and \(b = 3\). Plugging these values into the formula gives us:
\[ x = -\frac{3}{2 \cdot 4} = -\frac{3}{8} = -0.375 \]
Next, we need to find the y-coordinate of the vertex by substituting \(x = -0.375\) back into the original equation:
\[ y = 4(-0.375)^2 + 3(-0.375) - 8 \]
Calculating \(4(-0.375)^2\):
\[ 4 \cdot 0.140625 = 0.5625 \]
Calculating \(3(-0.375)\):
\[ -1.125 \]
Now putting it all together:
\[ y = 0.5625 - 1.125 - 8 = 0.5625 - 1.125 - 8 = -8.5625 \]
Thus, the vertex of the quadratic \(y = 4x^2 + 3x - 8\) is approximately \((-0.375, -8.563)\).
So, the correct response is:
\[ (-0.375, -8.563) \]