To find the vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \), we can use the vertex formula. The x-coordinate of the vertex for a quadratic in the form \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
Here, \( a = 4 \) and \( b = 3 \).
Substituting the values:
\[ x = -\frac{3}{2(4)} = -\frac{3}{8} = -0.375 \]
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 \( (-0.375)^2 \):
\[ (-0.375)^2 = 0.140625 \]
Now substituting this value into the equation:
\[ y = 4(0.140625) + 3(-0.375) - 8 \]
Calculating \( 4(0.140625) \):
\[ 4(0.140625) = 0.5625 \]
Calculating \( 3(-0.375) \):
\[ 3(-0.375) = -1.125 \]
Putting this all together:
\[ y = 0.5625 - 1.125 - 8 = 0.5625 - 1.125 - 8 = -8.5625 \]
So, the vertex of the quadratic \( y = 4x^2 + 3x - 8 \) is:
\[ (-0.375, -8.5625) \]
None of the given responses match the exact vertex calculated, but the closest option is:
(−0.375,−8.563)
Thus, the answer is \( (-0.375, -8.563) \).
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