Answer: The conic form equation for this parabola is (y - 4)^2 = (36 x +4)
Step-by-step explanation:
The equation \( (y - 4)^2 = 36(x + 4) \) represents a parabola. To write it in conic form, which is typically expressed as \( (x - h)^2 = 4p(y - k) \) for a parabola with a vertical axis of symmetry or \( (y - k)^2 = 4p(x - h) \) for a parabola with a horizontal axis of symmetry, you need to complete the square. Here, the axis of symmetry is vertical, so we'll use the form \( (x - h)^2 = 4p(y - k) \).
1. Start by expanding \( (y - 4)^2 \) on the left side of the equation:
\( (y - 4)(y - 4) = 36(x + 4) \)
2. Expand and simplify the left side:
\( y^2 - 8y + 16 = 36(x + 4) \)
3. Move the constant term (16) to the right side of the equation:
\( y^2 - 8y = 36(x + 4) - 16 \)
4. Factor a 4 out of the right side:
\( y^2 - 8y = 4(9(x + 4) - 4) \)
5. Further simplify the right side:
\( y^2 - 8y = 4(9x + 36 - 4) \)
\( y^2 - 8y = 4(9x + 32) \)
6. Complete the square on the left side. To do this, add and subtract \((8/2)^2 = 16\) inside the parentheses:
\( y^2 - 8y + 16 - 16 = 4(9x + 32) \)
7. Simplify the left side:
\( (y^2 - 8y + 16) - 16 = 4(9x + 32) \)
\( (y^2 - 8y + 16) = 4(9x + 32) \)
8. Recognize that the left side is a perfect square:
\( (y - 4)^2 = 4(9x + 32) \)
Now, the equation is in the conic form \( (y - k)^2 = 4p(x - h) \) for a parabola with its vertex at the point \((h, k)\). In this case, the vertex is \((-4, 4)\), and \(p = 9\). So, the equation is:
\( (y - 4)^2 = 4(9)(x - (-4)) \)
\( (y - 4)^2 = 36(x + 4) \)
The conic form equation for this parabola is \( (y - 4)^2 = 36(x + 4) \).