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Answer:
- II and III only
Step-by-step explanation:
[tex]\item \textbf{Line of Symmetry and Vertex Location:} \begin{itemize} \item The parabola passes through (0,0) and (6,0), which are the x-intercepts. \item The midpoint of the segment connecting these points is (3,0), so the line of symmetry is \( x = 3 \). \item Since the vertex \( T(h, 4) \) lies on the line of symmetry, \( h = 3 \). \end{itemize}[/tex]
[tex]\\\\\( h = 3 \), so Choice I (which states \( h = 2 \)) is false.[/tex]
[tex]\textbf{Symmetry and the Point (1,2):} \begin{itemize} \item If the parabola passes through \( (1,2) \), this point is 2 units to the left of the line of symmetry at \( x = 3 \). \item Due to symmetry, there must be a matching point 2 units to the right of \( x = 3 \), which is \( (5,2) \). \end{itemize}[/tex]
[tex]\\ \text{If the parabola passes through \( (1,2) \), it must also pass through \( (5,2) \), so Choice II is true.}[/tex]
[tex]\item \textbf{Vertex as the Highest or Lowest Point:}\\\begin{itemize} \item The vertex \( T(3,4) \) is either the highest or lowest point on the parabola.\\[/tex]
Since the parabola is facing downwards, the vertex (3,4) is the highest point.
Therefore, options II and III are correct.