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The question is asking how the graph of the quadratic function \( g(x) = x^2 + 4x + 10 \) compares to the graph of its parent function \( f(x) = x^2 \).

To analyze this, let's break it down:

### 1. Parent Function:
The parent function \( f(x) = x^2 \) is a basic parabola, opening upwards with its vertex at the origin \((0, 0)\).

### 2. Transformations of \( g(x) = x^2 + 4x + 10 \):
The function \( g(x) = x^2 + 4x + 10 \) can be rewritten in vertex form to better understand its graph. To do this, we complete the square:

- Start with \( g(x) = x^2 + 4x + 10 \).
- Complete the square by taking half of the coefficient of \( x \) (which is 4), squaring it, and adding and subtracting that value inside the equation:

\[
g(x) = (x^2 + 4x + 4) + 10 - 4 = (x + 2)^2 + 6.
\]

Now the function is in vertex form: \( g(x) = (x + 2)^2 + 6 \).

### 3. Comparison:
- **Shift:** The graph of \( g(x) \) is shifted **left by 2 units** and **up by 6 units** compared to the parent function \( f(x) = x^2 \).
- **Same Shape:** Since the coefficient of \( x^2 \) is still 1, the parabola opens upward and has the same width as the parent function.
- **Vertex:** The vertex of \( g(x) \) is at \( (-2, 6) \), while the vertex of \( f(x) \) is at \( (0, 0) \).

In summary, the graph of \( g(x) \) is the same shape as \( f(x) \), but shifted 2 units to the left and 6 units up.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How do you complete the square for other quadratic functions?
2. How do transformations affect the graph of any quadratic function?
3. What does it mean when the quadratic coefficient changes?
4. How can you determine the axis of symmetry from a quadratic equation?
5. How can we find the roots of a quadratic equation graphically and algebraically?

**Tip:** Completing the square is a powerful tool for rewriting quadratic functions in vertex form, making it easier to identify transformations such as shifts and stretches.

How does the graph of the quadratic function g(x) = x^2 + 4x + 10 compare to the graph of its parent function, f(x) = x^2?

Explore how the graph of the quadratic function g(x) = x^2 + 4x + 10 compares to its parent function f(x) = x^2. Learn about graph transformations such as shifts and vertex form, using techniques like completing the square. Suitable for Grades 9-12.

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