A parabola passes through the points (0, 0) and (6, 0). If the turning point of the parabola is `T(h, 4)`, which statement must be true?
I. h=2
II. If the parabola passes through (1, 2), then it must also pass through (5, 2).
III. Point T is the highest point of the parabola

Options

A. II only
B. III only
C. I and II only
D. II and III only

Answer

To solve the problem, we need to analyze the properties of the parabola given the points it passes through and its turning point.

### Step 1: Identify the roots of the parabola
The parabola passes through the points (0, 0) and (6, 0). These points are the roots of the parabola.

### Step 2: Determine the vertex (turning point) of the parabola
The vertex (turning point) of a parabola that opens downwards (since the turning point is at \(y = 4\)) lies halfway between the roots. The x-coordinate of the vertex can be calculated as:
\[ h = \frac{x_1 + x_2}{2} = \frac{0 + 6}{2} = 3 \]
Thus, the vertex is at the point \(T(3, 4)\).

### Step 3: Evaluate statement I: \(h = 2\)
From our calculation, we found that \(h = 3\). Therefore, statement I is false.

### Step 4: Evaluate statement II: If the parabola passes through (1, 2), then it must also pass through (5, 2)
The points (1, 2) and (5, 2) are equidistant from the vertex (3, 4). If the parabola passes through (1, 2), due to its symmetry, it must also pass through (5, 2). Thus, statement II is true.

### Step 5: Evaluate statement III: Point T is the highest point of the parabola
Since the parabola opens downwards (as inferred from the turning point being at \(y = 4\) and the roots being at \(y = 0\)), the vertex at \(T(3, 4)\) is indeed the highest point of the parabola. Therefore, statement III is true.

### Conclusion
Based on the evaluations:
- Statement I is false.
- Statement II is true.
- Statement III is true.

Thus, the correct answer is that statements II and III are true.