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You need to graph the function \( h(x) = \frac{x^2 + 8x + 16}{x} \).

### Step-by-step Breakdown:
1. **Simplify the expression**:
   The numerator can be factored:
   \[
   x^2 + 8x + 16 = (x + 4)^2
   \]
   So, the function becomes:
   \[
   h(x) = \frac{(x + 4)^2}{x}
   \]
   Which simplifies further to:
   \[
   h(x) = x + 8 + \frac{16}{x}
   \]
   This gives us three terms: \( x \), 8, and \( \frac{16}{x} \).

2. **Find any restrictions**:
   The function is undefined when \( x = 0 \), since division by zero is not allowed. This indicates a vertical asymptote at \( x = 0 \).

3. **Analyze the behavior of the function**:
   - For large positive and large negative values of \( x \), the \( \frac{16}{x} \) term approaches zero, so \( h(x) \) behaves like \( x + 8 \).
   - As \( x \to 0 \) from the positive side, \( h(x) \) grows very large (positive).
   - As \( x \to 0 \) from the negative side, \( h(x) \) becomes very negative.

4. **Asymptotes**:
   - Vertical asymptote at \( x = 0 \).
   - No horizontal asymptote, but the function behaves like \( x + 8 \) for large \( |x| \).

### Now, let's describe the graph:
- **Vertical asymptote** at \( x = 0 \).
- **Parabolic-like behavior** on both sides, but approaching the line \( y = x + 8 \) as \( x \to \infty \).
- At large positive values of \( x \), the graph closely follows \( y = x + 8 \).
- For negative \( x \), it dips down dramatically as \( x \) approaches 0 from the left.

You can now graph the function based on this behavior!

Would you like me to show a more detailed explanation of any specific part? Here are some additional questions you might find useful:

1. How do vertical asymptotes affect graph behavior?
2. What happens to \( h(x) \) as \( x \to 0 \) from the left and right?
3. Can we approximate \( h(x) \) for very large \( |x| \)?
4. How does factoring help simplify the function?
5. Why does \( x = 0 \) cause a discontinuity?

**Tip**: Always check for discontinuities and asymptotes when graphing rational functions.

Graph the function h(x) = (x^2 + 8x + 16) / x

Learn how to graph the rational function h(x) = (x^2 + 8x + 16) / x. This problem includes factoring the quadratic expression, analyzing vertical asymptotes, and understanding the behavior of the function at discontinuities. Suitable for students in Grades 10-12.

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