Write the General Form of a Parabola in Standard Form: Find the Vertex, Focus, and Directrix (Left)

The video begins by introducing the task: finding the vertex, focus, and directrix for the parabola given by the equation y^2 - 6y + 8x + 41 = 0. It explains that to do this, the equation must be written in standard form. Since the equation contains y squared, it will open either left or right, corresponding to the form (y-k)^2 = 4p(x-h).

The first step in converting to standard form is to isolate the y terms on the left side of the equation and move other terms to the right. This involves subtracting 8x and 41 from both sides, resulting in y^2 - 6y = -8x - 41.

To complete the square on the left side, the video explains adding the square of half the coefficient of y to both sides. Half of -6 is -3, and (-3)^2 is 9. Adding 9 to both sides gives y^2 - 6y + 9 = -8x - 41 + 9.

The left side (y^2 - 6y + 9) factors into (y-3)^2. The right side simplifies to -8x - 32. Factoring out -8 from the right side yields (y-3)^2 = -8(x+4). This is the standard form of the parabola's equation.

From the standard form, 4p = -8, so p = -2. Since p is negative and the y part is squared, the parabola opens to the left.

The vertex (h, k) is determined from the standard form. Since we have (x+4), h is -4, and since we have (y-3), k is 3. Thus, the vertex is at (-4, 3).

Because the parabola opens left, the focus is to the left of the vertex. The distance from the vertex to the focus is |p| units, which is 2 units. To find the focus, subtract 2 from the x-coordinate of the vertex: (-4 - 2, 3), resulting in the focus at (-6, 3).

The directrix is a vertical line located |p| units to the right of the vertex. Since the vertex's x-coordinate is -4, add 2 to it: x = -4 + 2. Therefore, the equation of the directrix is x = -2.

A final sketch of the parabola is provided, showing it opening to the left, with the vertex, focus, and directrix in their correct positions relative to each other.