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!