Questions: Find the vertex, focus, and directrix of the parabola. Graph the parabola. x^2 + 18x = 9y - 18 The vertex is (-9,-7). (Type an ordered pair. Simplify your answer.) The focus is . (Type an ordered pair. Simplify your answer.)

Solution Steps

Given the equation of the parabola: \[ x^{2} + 18x = 9y - 18 \] First, we rewrite it in the standard form of a parabola. We complete the square for the \(x\) terms.

\[ x^{2} + 18x = 9y - 18 \] \[ x^{2} + 18x + 81 = 9y - 18 + 81 \] \[ (x + 9)^{2} = 9y + 63 \] \[ (x + 9)^{2} = 9(y + 7) \]

The standard form of a parabola that opens upwards or downwards is: \[ (x - h)^{2} = 4p(y - k) \] Comparing \((x + 9)^{2} = 9(y + 7)\) with \((x - h)^{2} = 4p(y - k)\), we get: \[ h = -9, \quad k = -7, \quad 4p = 9 \] Thus, the vertex is \((-9, -7)\).

From \(4p = 9\), we find \(p\): \[ p = \frac{9}{4} = 2.25 \] Since the parabola opens upwards, the focus is \(p\) units above the vertex: \[ \text{Focus} = (-9, -7 + 2.25) = (-9, -4.75) \]

Final Answer