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The graph of h(x) = g(x + 5) has a. the same horizontal asymptote as function g a. the same horizontal asymptote as function g b. a horizontal asymptote at y=5 c. a horizontal asymptote at y=8 and a. the same vertical asymptote as function g b. a vertical asymptote at x=-7 c. a vertical asymptote at x=-5 d. a vertical asymptote at x=3 pick the correct answers fill in the blank

To answer the question regarding the graph of \( h(x) = g(x + 5) \), let's analyze the effects of the transformation on the graph of \( g(x) \).

### Horizontal Asymptote
The horizontal asymptote of a function is affected by vertical shifts but not by horizontal shifts. Since \( h(x) = g(x + 5) \) represents a horizontal shift to the left by 5 units, the horizontal asymptote remains unchanged. 

From the given graph, the horizontal asymptote of \( g(x) \) appears to be \( y = 0 \). Thus, the horizontal asymptote of \( h(x) \) is also \( y = 0 \).

### Vertical Asymptote
Vertical asymptotes of a function are affected by horizontal shifts. If \( g(x) \) has a vertical asymptote at \( x = a \), then \( h(x) = g(x + 5) \) will have a vertical asymptote at \( x = a - 5 \).

From the graph, \( g(x) \) has a vertical asymptote at \( x = 0 \). Shifting this vertical asymptote to the left by 5 units gives us a new vertical asymptote at \( x = -5 \).

### Conclusion
So, the graph of \( h(x) = g(x + 5) \) has:
- The same horizontal asymptote as function \( g \).
- A vertical asymptote at \( x = -5 \).

Thus, the correct answers to the blanks are:
- **the same horizontal asymptote as function g**
- **a vertical asymptote at x=-5**

If you have any questions or need further details, feel free to ask!

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Here are 8 related questions for further practice:
1. What is the effect of a vertical shift on the horizontal asymptote of a rational function?
2. How do you determine the horizontal asymptote of a rational function?
3. What happens to the vertical asymptote of \( g(x) \) if you consider \( h(x) = g(x - 2) \)?
4. How would the graph change if you consider \( h(x) = g(2x) \)?
5. What is the difference between a horizontal and a vertical shift in terms of function transformations?
6. How does adding a constant inside the argument of a function affect its graph?
7. Can a rational function have more than one horizontal asymptote? Why or why not?
8. How do you find the vertical asymptotes of a rational function algebraically?

**Tip:** Always plot a few points and the asymptotes when sketching a transformed function to better understand the shifts and transformations.

Explore how the graph of h(x) = g(x + 5) relates to the original function g(x), including its horizontal and vertical asymptotes. This detailed explanation covers function transformations, horizontal asymptotes, and vertical shifts. Suitable for high school students.

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