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The equation given is \( y = (x - 3)^2 + 1 \). This is the equation of a parabola in vertex form, \( y = a(x - h)^2 + k \), where the vertex is at the point \( (h, k) \).

In this case, the vertex is at \( (3, 1) \). The parabola opens upwards because the coefficient of the squared term is positive (i.e., 1). Now, let's find the points to graph the parabola:

1. **Vertex**: \( (3, 1) \)

2. **Two points to the left of the vertex**:
   - For \( x = 2 \): \( y = (2 - 3)^2 + 1 = 1^2 + 1 = 2 \), so \( (2, 2) \)
   - For \( x = 1 \): \( y = (1 - 3)^2 + 1 = 2^2 + 1 = 5 \), so \( (1, 5) \)

3. **Two points to the right of the vertex**:
   - For \( x = 4 \): \( y = (4 - 3)^2 + 1 = 1^2 + 1 = 2 \), so \( (4, 2) \)
   - For \( x = 5 \): \( y = (5 - 3)^2 + 1 = 2^2 + 1 = 5 \), so \( (5, 5) \)

So, the five points to plot on the graph are:
- \( (3, 1) \) (vertex)
- \( (2, 2) \)
- \( (1, 5) \)
- \( (4, 2) \)
- \( (5, 5) \)

Would you like further clarification or a graph of this parabola?

Here are five related questions to deepen understanding:
1. What is the general form of a quadratic equation?
2. How can we determine whether a parabola opens upwards or downwards?
3. How does changing the value of \( h \) in the vertex form affect the graph?
4. What is the axis of symmetry for this parabola?
5. How does the value of \( k \) affect the position of the parabola?

**Tip:** The vertex form \( y = a(x-h)^2 + k \) makes it easy to identify the vertex and understand transformations like shifting.

Graph the parabola y = (x - 3)^2 + 1. Plot five points: the vertex, two points to the left of the vertex, and two points to the right.

Learn how to graph the quadratic equation y = (x - 3)^2 + 1 by plotting the vertex and points to the left and right of it. Understand the vertex form and apply it to graph parabolas easily. Suitable for students in Grades 8-10.

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