Answer :
To determine the vertex, focus, and directrix of the parabola given by the equation [tex]\( x^2 + 8x + 4y + 4 = 0 \)[/tex], we proceed with the following steps:
1. Rearrange the given equation: Start with the equation:
[tex]\[ x^2 + 8x + 4y + 4 = 0 \][/tex]
Rearrange it to isolate the [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 + 8x + 4y = -4 \][/tex]
2. Complete the square for the [tex]\( x \)[/tex]-terms. To complete the square for [tex]\( x^2 + 8x \)[/tex]:
- Take half of the coefficient of [tex]\( x \)[/tex], which is 8, divide it by 2 to get 4, and then square it to get 16.
- Add and subtract 16 inside the equation:
[tex]\[ x^2 + 8x + 16 - 16 + 4y = -4 \][/tex]
- Rewrite the trinomial [tex]\( x^2 + 8x + 16 \)[/tex] as a square and simplify:
[tex]\[ (x+4)^2 - 16 + 4y = -4 \][/tex]
3. Simplify the equation: Move the constants to the right side:
[tex]\[ (x+4)^2 - 16 + 4y = -4 \][/tex]
[tex]\[ (x+4)^2 + 4y = 12 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ (x+4)^2 = -4(y + 3) \][/tex]
4. Identify the vertex: The standard form of a vertical parabola is [tex]\( (x - h)^2 = 4p(y - k) \)[/tex]. For this equation, we have:
[tex]\[ (x + 4)^2 = -4(y + 3) \][/tex]
Comparing this with [tex]\( (x - h)^2 = 4p(y - k) \)[/tex]:
- The vertex [tex]\((h, k)\)[/tex] is [tex]\((-4, -3)\)[/tex].
5. Determine [tex]\( p \)[/tex]: In the equation:
[tex]\[ 4p = -4 \implies p = -1 \][/tex]
6. Find the focus and directrix:
- The focus is located at [tex]\((h, k + p)\)[/tex]. For our vertex [tex]\((-4, -3)\)[/tex] and [tex]\( p = -1 \)[/tex]:
[tex]\[ \text{Focus} = (-4, -3 + (-1)) = (-4, -4) \][/tex]
- The directrix is [tex]\( y = k - p \)[/tex]. For our vertex [tex]\((-4, -3)\)[/tex] and [tex]\( p = -1 \)[/tex]:
[tex]\[ \text{Directrix} = y = -3 - (-1) = -3 + 1 = -2 \][/tex]
Putting all the information together, we have:
- Vertex: [tex]\((-4, -3)\)[/tex]
- Focus: [tex]\((-4, -4)\)[/tex]
- Directrix: [tex]\(y = -2\)[/tex]
Thus, the correct answer is:
vertex [tex]\((-4, -3)\)[/tex]; focus [tex]\((-4, -4)\)[/tex]; directrix [tex]\(y = -2\)[/tex].
1. Rearrange the given equation: Start with the equation:
[tex]\[ x^2 + 8x + 4y + 4 = 0 \][/tex]
Rearrange it to isolate the [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 + 8x + 4y = -4 \][/tex]
2. Complete the square for the [tex]\( x \)[/tex]-terms. To complete the square for [tex]\( x^2 + 8x \)[/tex]:
- Take half of the coefficient of [tex]\( x \)[/tex], which is 8, divide it by 2 to get 4, and then square it to get 16.
- Add and subtract 16 inside the equation:
[tex]\[ x^2 + 8x + 16 - 16 + 4y = -4 \][/tex]
- Rewrite the trinomial [tex]\( x^2 + 8x + 16 \)[/tex] as a square and simplify:
[tex]\[ (x+4)^2 - 16 + 4y = -4 \][/tex]
3. Simplify the equation: Move the constants to the right side:
[tex]\[ (x+4)^2 - 16 + 4y = -4 \][/tex]
[tex]\[ (x+4)^2 + 4y = 12 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ (x+4)^2 = -4(y + 3) \][/tex]
4. Identify the vertex: The standard form of a vertical parabola is [tex]\( (x - h)^2 = 4p(y - k) \)[/tex]. For this equation, we have:
[tex]\[ (x + 4)^2 = -4(y + 3) \][/tex]
Comparing this with [tex]\( (x - h)^2 = 4p(y - k) \)[/tex]:
- The vertex [tex]\((h, k)\)[/tex] is [tex]\((-4, -3)\)[/tex].
5. Determine [tex]\( p \)[/tex]: In the equation:
[tex]\[ 4p = -4 \implies p = -1 \][/tex]
6. Find the focus and directrix:
- The focus is located at [tex]\((h, k + p)\)[/tex]. For our vertex [tex]\((-4, -3)\)[/tex] and [tex]\( p = -1 \)[/tex]:
[tex]\[ \text{Focus} = (-4, -3 + (-1)) = (-4, -4) \][/tex]
- The directrix is [tex]\( y = k - p \)[/tex]. For our vertex [tex]\((-4, -3)\)[/tex] and [tex]\( p = -1 \)[/tex]:
[tex]\[ \text{Directrix} = y = -3 - (-1) = -3 + 1 = -2 \][/tex]
Putting all the information together, we have:
- Vertex: [tex]\((-4, -3)\)[/tex]
- Focus: [tex]\((-4, -4)\)[/tex]
- Directrix: [tex]\(y = -2\)[/tex]
Thus, the correct answer is:
vertex [tex]\((-4, -3)\)[/tex]; focus [tex]\((-4, -4)\)[/tex]; directrix [tex]\(y = -2\)[/tex].
