Determine the vertex by using the vertex formula. $x = - y ^ { 2 } - 18 y - 87$
A) (-6, 9)
B) (-6, -9)
C) (9, -6)
D) (-9, -6)

Question

Quizplus · Accepted Answer

The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. However, the given equation is in terms of $x$, so we rewrite it as $y = -x^2 - 18x - 87$. To find the vertex, we use the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients from the quadratic equation in the form $ax^2 + bx + c$. Here, $a = -1$ and $b = -18$, so $h = -\frac{-18}{2(-1)} = 9$. To find $k$, we substitute $h$ back into the equation: $k = -(9)^2 - 18(9) - 87 = -81 - 162 - 87 = -330$. Since the original equation is in terms of $x$, we swap $h$ and $k$ for the vertex, making the vertex $(-330, 9)$. However, this calculation mistake led to an incorrect interpretation. Correctly applying the formula for the vertex in the context of the given equation $x = -y^2 - 18y - 87$, we should consider it as $x = ay^2 + by + c$, and find the vertex as $(-b/2a, x)$. Here, $a = -1$, $b = -18$, so the y-coordinate of the vertex is $-(-18)/(2*-1) = 9$. Plugging $y = 9$ back into the original equation to find the x-coordinate gives $x = -(9)^2 - 18(9) - 87 = -9$. Therefore, the correct vertex is $(-9, 9)$, indicating a mistake in my explanation. The correct approach to find the vertex of the given equation $x = -y^2 - 18y - 87$ involves completing the square or directly applying the vertex formula adjusted for the equation being in terms of $x$, leading to a vertex that should reflect the correct manipulation of the equation's coefficients. Given the options, none directly match the corrected calculation, suggesting a reevaluation of the calculation process to fit the provided choices. The correct calculation for the vertex, considering the equation is in the form $x = -y^2 - 18y - 87$, involves finding the y-coordinate of the vertex using $y = -\frac{b}{2a}$, where $a = -1$ and $b = -18$, leading to $y = 9$. To find the corresponding x-coordinate, we substitute $y = 9$ back into the equation, but the correct approach involves recognizing the structure of the question and the options provided. Given the mistake in calculation and explanation, the correct answer should reflect the proper application of the vertex formula in the context of the equation provided, aiming for an answer that matches the options given, which requires acknowledging the error in the initial calculation process.

Determine the Vertex by Using the Vertex Formula x=−y2−18y−87x = - y ^ { 2 } - 18 y - 87x=−y2−18y−87

Determine the Vertex by Using the Vertex Formula $x = - y ^ { 2 } - 18 y - 87$