To find the vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \), we can use the vertex formula \( x = -\frac{b}{2a} \), where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \).
In this equation:
- \( a = 4 \)
- \( b = 3 \)
Now, we can calculate the \( x \)-coordinate of the vertex:
\[ x = -\frac{b}{2a} = -\frac{3}{2 \cdot 4} = -\frac{3}{8} = -0.375 \]
Next, we need to find the \( y \)-coordinate by substituting \( x = -0.375 \) back into the quadratic equation:
\[ y = 4(-0.375)^2 + 3(-0.375) - 8 \]
Calculating \( (-0.375)^2 \):
\[ (-0.375)^2 = 0.140625 \]
Now substituting this back in:
\[ y = 4(0.140625) + 3(-0.375) - 8 \] \[ y = 0.5625 - 1.125 - 8 \] \[ y = 0.5625 - 1.125 - 8 = -8.5625 \]
Thus, the vertex of the quadratic is:
\[ (-0.375, -8.5625) \]
The closest match from your provided options is:
(−0.375, −8.563)
So the vertex of the quadratic is \((-0.375, -8.563)\).
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