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 \( y = ax^2 + bx + c \):
\[ x = -\frac{b}{2a} \]
Here, \( a = 4 \) and \( b = 3 \).
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{3}{2 \cdot 4} = -\frac{3}{8} = -0.375 \]
Now, to find the y-coordinate of the vertex, we substitute \( x = -0.375 \) back into the quadratic equation:
\[ y = 4(-0.375)^2 + 3(-0.375) - 8 \]
Calculating this step-by-step:
- Calculate \( (-0.375)^2 = 0.140625 \)
- Multiply by 4: \( 4 \cdot 0.140625 = 0.5625 \)
- Calculate \( 3 \cdot (-0.375) = -1.125 \)
- Now, combine:
\[ y = 0.5625 - 1.125 - 8 = 0.5625 - 1.125 - 8 = -0.5625 - 8 = -8.5625 \]
Thus, the vertex of the quadratic \( y = 4x^2 + 3x - 8 \) is:
\[ (-0.375, -8.5625) \]
Since the option closest to this value in your responses is:
(−0.375,−8.563)
So the answer is:
(−0.375,−8.563).