To find the vertex of the quadratic function \( y = 4x^2 + 3x - 8 \), we can use the vertex formula. The x-coordinate of the vertex for a quadratic in the form \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In this equation:
- \( a = 4 \)
- \( b = 3 \)
Substituting these values into the formula gives us:
\[ x = -\frac{3}{2 \times 4} = -\frac{3}{8} = -0.375 \]
Next, we substitute this x-value back into the original equation to find the y-coordinate of the vertex:
\[ y = 4(-0.375)^2 + 3(-0.375) - 8 \]
Calculating each term step-by-step:
- Calculate \( (-0.375)^2 = 0.140625 \).
- Calculate \( 4 \times 0.140625 = 0.5625 \).
- Calculate \( 3 \times (-0.375) = -1.125 \).
- Now substitute back into the equation:
\[ y = 0.5625 - 1.125 - 8 = 0.5625 - 1.125 = -0.5625 \quad \text{and then} \quad -0.5625 - 8 = -8.5625 \]
So the vertex is:
\[ (-0.375, -8.5625) \]
Considering the available responses and rounding:
The approximate coordinates of the vertex are:
\((-0.375, -8.563)\).
Thus, the correct answer is:
(−0.375,−8.563)