https://file.aiquickdraw.com/mathbot/file/1728759375289cs5bnpvh.jpg

To solve this problem, let's analyze the given quadratic function:

### Given Function:
\[
h(x) = (x - 3)^2 + 2
\]

### Comparison Function:
\[
y = x^2
\]

### Step 1: Understand the General Form
The function \( h(x) = (x - 3)^2 + 2 \) is a transformation of the standard quadratic function \( y = x^2 \). This is in the **vertex form** of a parabola:
\[
y = (x - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola.

- In this case, the vertex is at \( (3, 2) \).
- The graph of \( y = x^2 \) has a vertex at \( (0, 0) \), which is the origin.

### Step 2: Transformations
The transformations that have been applied to \( y = x^2 \) are:
- **Horizontal shift**: The graph is shifted 3 units to the right, because of \( (x - 3) \).
- **Vertical shift**: The graph is shifted 2 units up, because of the \( +2 \).

### Step 3: Graph the Function
1. The graph of \( h(x) = (x - 3)^2 + 2 \) will have the same "U" shape as \( y = x^2 \) but will be:
   - Shifted to the right by 3 units.
   - Shifted up by 2 units.

### Step 4: Comparison with \( y = x^2 \)
- **Shape**: Both graphs are parabolas with the same shape since the coefficient of \( x^2 \) is the same (no stretching or compressing).
- **Position**: \( h(x) \) is translated 3 units to the right and 2 units up compared to \( y = x^2 \).

Would you like me to generate a visual graph for this? 

---

Let me know if you have any questions or need further details. Here are some related questions you might explore:

1. How would the graph change if we replaced \( +2 \) with \( -2 \)?
2. What if the function was \( h(x) = 2(x-3)^2 + 2 \)? How would this affect the shape?
3. What is the axis of symmetry for both functions?
4. How do you find the y-intercept for \( h(x) \)?
5. Can you find the x-intercepts (roots) of the function \( h(x) \)?

**Tip**: Understanding transformations in vertex form helps visualize changes without plotting the full graph!

Sketch the graph of the quadratic function h(x) = (x - 3)^2 + 2 and compare it with the graph of y = x^2.

Explore how to graph the quadratic function h(x) = (x - 3)^2 + 2 and compare it with y = x^2. Learn about key graph transformations, vertex form, and translations of parabolas. Suitable for students in Grades 8-10.

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