The given equation is:

$(x - 3)^2 = 4(y - 3)$

This is the standard form of a parabola equation. To interpret it:

• The equation $(x - h)^2 = 4p(y - k)$ represents a parabola that opens either upwards or downwards.
• In this case, the vertex is at the point $(h, k)$.

Step-by-step explanation:

1. The equation is already in the form $(x - h)^2 = 4p(y - k)$, where:

   • $h = 3$
   • $k = 3$
   • $4p = 4$ implies that $p = 1$

2. This indicates:

   • The vertex of the parabola is at $(3, 3)$.
   • The parabola opens upwards since $p > 0$.
   • The focal length (the distance from the vertex to the focus) is $p = 1$.

Parabola features:

• Vertex: $(3, 3)$
• Focus: Since $p = 1$ and the parabola opens upward, the focus is at $(3, 3 + 1) = (3, 4)$.
• Directrix: The directrix is a horizontal line below the vertex, given by $y = 3 - 1 = 2$.
• Axis of symmetry: The axis of symmetry is the vertical line $x = 3$.

Would you like more details on the graph or specific properties of this parabola?

Here are 5 related questions:

1. How do we find the equation of a parabola given the vertex and focus?
2. What is the general form of the parabola equation?
3. How do we calculate the latus rectum length in a parabola?
4. What are the properties of a parabola opening downward?
5. Can the same process be applied to find the directrix for horizontal parabolas?

Tip: For parabolas, always identify the vertex and direction of opening before proceeding to focus or directrix calculations.