Let's go through each of these problems step-by-step.

1. Trace the parabola with the equation $y^2 - 4x - 12y + 28 = 0$

To trace this parabola, let's rewrite it in a standard form by completing the square:

1. Start with the equation: $y^2 - 12y = 4x - 28$.
2. Complete the square for the $y$-terms:
   • $y^2 - 12y$ becomes $(y - 6)^2 - 36$.
   • Substitute back: $(y - 6)^2 - 36 = 4x - 28$.
3. Rewrite it: $(y - 6)^2 = 4x + 8$.
4. Simplify further to get: $(y - 6)^2 = 4(x + 2)$.

This is the equation of a parabola that opens to the right, with vertex at $(-2, 6)$ and focus shifted based on the coefficient of $x$.

2. Find the equation of the parabola with vertex on the y-axis, axis parallel to the x-axis, and passing through $(2, 2)$ and $(8, -1)$

For a parabola with a vertex on the y-axis and axis parallel to the x-axis, the general form would be $y = ax^2 + bx + c$ or a form of $(x - h)^2 = 4p(y - k)$, where $h = 0$.

Using the points provided, we can set up equations and solve for the parameters $a$, $b$, $c$, or $p$.

3. Find the equation of the parabola with vertex on the line $y = 2$, axis parallel to Oy, length of the latus rectum 6, and passing through $(2, 8)$

Since the parabola has its vertex on the line $y = 2$, it’s vertically oriented. The length of the latus rectum being 6 implies $4p = 6$ or $p = 1.5$.

The standard form for a vertically oriented parabola with vertex $(h, k)$ is $(x - h)^2 = 4p(y - k)$.

4. Find the equation of a parabola with axis parallel to $Ox$, and passing through $(5, 4)$, $(11, 2)$, and $(21, -4)$

Here, we would use the general form $y = ax^2 + bx + c$ and plug in the three points to create a system of equations to solve for $a$, $b$, and $c$.

5. Find the equation of a parabola with vertex on the line $y = x + 2$, axis parallel to Oy, latus rectum 6, and passing through $(-3, -1)$

For a parabola with a vertex on a slant line, further transformation is needed. The form can still be based on the latus rectum, but the placement of the vertex and orientation of the parabola would depend on the line's slope and other constraints.

Would you like detailed steps for each equation, or is there a specific one you want more help with?

Here are some related questions:

1. How do you complete the square for different types of equations?
2. What is the significance of the latus rectum in parabola equations?
3. How can you determine the direction a parabola opens based on its equation?
4. What are the steps to solve a system of equations with multiple unknowns?
5. How can transformations be applied to shift the vertex of a parabola?

Tip: Always start by identifying the orientation and vertex position for conic sections like parabolas to simplify equation transformations.