Questions: Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x, y) point. y=4 x^2+56 x+188

Solution Steps

To find the coordinates of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \), we use the vertex formula. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Once we have the x-coordinate, we substitute it back into the equation to find the y-coordinate.

The given quadratic equation is \( y = 4x^2 + 56x + 188 \). Here, the coefficients are:

• \( a = 4 \)
• \( b = 56 \)
• \( c = 188 \)

Using the vertex formula \( x = -\frac{b}{2a} \): \[ x = -\frac{56}{2 \cdot 4} = -\frac{56}{8} = -7.0 \]

Substituting \( x = -7.0 \) back into the equation to find \( y \): \[ y = 4(-7.0)^2 + 56(-7.0) + 188 \] Calculating each term: \[ y = 4(49) - 392 + 188 = 196 - 392 + 188 = -8.0 \]

Final Answer