Problem

Find the vertex, focus, and directrix of the parabola. Graph the parabola. \[ x^{2}+2 x=-y-4 \] The vertex is $(-1,-3)$. (Type an ordered pair.) The focus is $\square$. (Type an ordered pair.)

Solution

Step 1 :Rewrite the given equation $x^2+2x=-y-4$ as $x^2+2x+1=y+3$ by adding $1$ to both sides.

Step 2 :This can be further simplified to $(x+1)^2=y+3$.

Step 3 :The vertex of the parabola is $(-1,-3)$.

Step 4 :The equation $(x+1)^2=y+3$ is in the form $(x-h)^2=4p(y-k)$, so $4p=1$. Therefore, $p=\frac{1}{4}$.

Step 5 :The focus of the parabola is $(h,k+p)$, which is $(-1,-3+\frac{1}{4})=(-1,-\frac{11}{4})$.

Step 6 :The directrix of the parabola is $y=k-p$, which is $y=-3-\frac{1}{4}=y=-\frac{13}{4}$.

Step 7 :\(\boxed{\text{So, the vertex is } (-1,-3), \text{ the focus is } (-1,-\frac{11}{4}), \text{ and the directrix is } y=-\frac{13}{4}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/X3u6q3rFUC/

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