Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an left parenthesis, x, comma, y, right parenthesis(x,y) point.

Question

y, equals, 2, x, squared, plus, 28, x, plus, 116
y=2x
2
+28x+116

Answer 1

To find the coordinates of the vertex of the parabola given by the equation \( y = 2x^2 + 28x + 116 \), we can use the formula for the vertex of a parabola in standard form \( y = ax^2 + bx + c \).

The x-coordinate of the vertex can be found using the formula:

\[ x = -\frac{b}{2a} \]

Here, \( a = 2 \) and \( b = 28 \).

Substituting in these values:

\[ x = -\frac{28}{2 \times 2} = -\frac{28}{4} = -7 \]

Next, we find the y-coordinate by substituting \( x = -7 \) back into the original equation:

\[ y = 2(-7)^2 + 28(-7) + 116 \]

Calculating each term:

\[ y = 2(49) + 28(-7) + 116 \ y = 98 - 196 + 116 \ y = 98 - 196 + 116 = 18 \]

Thus, the coordinates of the vertex of the parabola are \( (-7, 18) \).

Therefore, the answer is:

\[ \boxed{(-7, 18)} \]