Geometry

Question Analysis  

The main focus is to determine the equation of a parabola using the given vertex, parameter , and a point, then identify its domain and range based on orientation.  

Key Concept Explanation  

Vertex Form of a Parabola:  

Vertical:  (opens up/down, ).  

Horizontal:  (opens left/right, ).  

Domain/Range:  

Vertical parabola: Domain = , Range depends on  and .  

Horizontal parabola: Range = , Domain depends on  and .  

Step-by-Step Solution  

1. Determine orientation (vertical/horizontal):  

Assume vertical parabola first (common unless specified otherwise):  

Test if the point  lies on it:  

.

Thus, it is a vertical parabola.  

2. Write the equation:  

3. Find domain and range:  

Domain: All real numbers () for vertical parabolas.  

Range: Since a = 3 > 0 (opens upward), the range starts at :  

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