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Let's analyze the given quadratic function: 

\[
f(x) = 5(x - 3)^2 + 2
\]

This is a quadratic function in vertex form, \( f(x) = a(x-h)^2 + k \), where:
- \( a = 5 \) (the coefficient in front of the squared term),
- \( h = 3 \) (the horizontal shift),
- \( k = 2 \) (the vertical shift).

### Key characteristics of the graph:
1. **Opens Upward or Downward**:
   - Since \( a = 5 \), which is positive, the parabola opens **upward**.

2. **Vertex**:
   - The vertex is at the point \( (h, k) = (3, 2) \).

3. **Maximum or Minimum**:
   - Because the parabola opens upward, it has a **minimum** value at the vertex, not a maximum.

Thus, the graph opens upward and has a minimum value at \( (3, 2) \).

### The correct answer:
"Opens upward and has minimum value at (3, 2)."

Would you like further details on this explanation or any other aspects? Here are some questions you might consider:

1. How do we derive the vertex form of a quadratic function?
2. Why does the sign of \(a\) determine whether the parabola opens upward or downward?
3. How do we calculate the axis of symmetry from the vertex form?
4. What does changing the value of \( k \) (vertical shift) do to the graph of the function?
5. How would the graph change if \( a \) were negative?

**Tip**: In any quadratic equation, the vertex form \( f(x) = a(x-h)^2 + k \) is very useful for quickly identifying the direction of the graph and the vertex point.

Describe the graph of the quadratic function: f(x) = 5(x - 3)^2 + 2.

Learn how to describe the graph of the quadratic function f(x) = 5(x - 3)^2 + 2. This problem covers key concepts of graphing quadratic functions, identifying the vertex, and understanding the direction of the parabola. Suitable for students in Grades 8-10.

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