The graph of `y=ax^(2)+bx+c` passes through the points `(1,-8), (2,-1), (3, 4), and (5, 8)`. If the maximum value of y-coordinate at x=5, through which outer other point must be graph of y pass?

Options

A. `(4, 6)`
B. `(6, 4)`
C. `(8, -1)`
D. `(10, -8)`

Explanation

To solve the problem step by step, we will analyze the given points, understand the properties of the quadratic function, and find the required point through which the graph must pass.

### Step 1: Identify the Vertex
The problem states that the maximum value of the y-coordinate occurs at \( x = 5 \). This indicates that the vertex of the parabola is at the point \( (5, 8) \).

**Hint:** The vertex of a parabola represented by \( y = ax^2 + bx + c \) is the point where the maximum or minimum value occurs.

### Step 2: Determine the Axis of Symmetry
For a parabola, the axis of symmetry can be found using the x-coordinate of the vertex. The axis of symmetry is the vertical line that passes through the vertex. Thus, the axis of symmetry is \( x = 5 \).

**Hint:** The axis of symmetry divides the parabola into two mirror-image halves.

### Step 3: Analyze the Given Points
The graph passes through the points \( (1, -8) \), \( (2, -1) \), \( (3, 4) \), and \( (5, 8) \). We will use the symmetry of the parabola to find the corresponding points on the left side of the axis of symmetry.

**Hint:** For every point \( (x, y) \) on one side of the axis of symmetry, there is a corresponding point \( (10 - x, y) \) on the other side.

### Step 4: Find the Symmetric Points
1. **For the point \( (1, -8) \)**:
- The distance from the axis \( x = 5 \) is \( 5 - 1 = 4 \).
- The symmetric point will be \( (5 + 4, -8) = (9, -8) \).

2. **For the point \( (2, -1) \)**:
- The distance from the axis \( x = 5 \) is \( 5 - 2 = 3 \).
- The symmetric point will be \( (5 + 3, -1) = (8, -1) \).

3. **For the point \( (3, 4) \)**:
- The distance from the axis \( x = 5 \) is \( 5 - 3 = 2 \).
- The symmetric point will be \( (5 + 2, 4) = (7, 4) \).

### Step 5: Conclusion
The graph of the quadratic equation must also pass through the point \( (9, -8) \) and \( (8, -1) \) as symmetric points. Since the question asks for "another point," we can conclude that the graph must pass through \( (8, -1) \).

**Final Answer:** The graph must also pass through the point \( (8, -1) \).