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Let 𝑦=𝑓(𝑥) be the function whose graph is given in the upper-left in the figure. Find formulas, in terms of 𝑓(𝑥) , for the horizontal and vertical shifts of the graph of 𝑓(𝑥) in parts (a)-(c) and determine the equation of each asymptote. Note only vertical and horizontal shifts have been applied to the graph of 𝑓 (no reflections, compressions, or stretches), so your formulas should be of the form 𝑦=𝑓(𝑥−ℎ)+𝑘 where ℎ and 𝑘 are constants you determine from each graph. (c) Using shifts of 𝑓(𝑥), graph C can be written as: y=? with an asymptote at y=?

To determine the transformation and equations associated with each graph, let's analyze the shifts applied to the base function \( f(x) \).

### Analysis of Graph C
The original graph of \( f(x) \) has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 5 \). The second image, representing graph C, shows a function that looks horizontally and vertically shifted.

Let's denote the transformations applied to \( f(x) \) in graph C as:

\[
y = f(x - h) + k
\]

where:
- \( h \) represents the horizontal shift.
- \( k \) represents the vertical shift.

### Step-by-Step Solution

1. **Identify Horizontal Shift**: Compare the location of the vertical asymptote in both graphs. For \( f(x) \), the vertical asymptote is at \( x = 0 \). In graph C, it is at \( x = 4 \). This indicates a horizontal shift of \( h = 4 \) units to the right.

2. **Identify Vertical Shift**: Compare the location of the horizontal asymptote in both graphs. For \( f(x) \), the horizontal asymptote is at \( y = 5 \). In graph C, the horizontal asymptote appears to be at \( y = 0 \). This indicates a vertical shift of \( k = -5 \).

Thus, the function for graph C, using shifts of \( f(x) \), can be written as:
\[
y = f(x - 4) - 5
\]

### Asymptotes for Graph C
- **Vertical Asymptote**: Since there was a horizontal shift of 4, the vertical asymptote moves from \( x = 0 \) to \( x = 4 \).
- **Horizontal Asymptote**: The horizontal asymptote for \( f(x) \) was \( y = 5 \). After shifting down by 5 units, it becomes \( y = 0 \).

### Final Answer
For graph C:
\[
y = f(x - 4) - 5
\]
with an asymptote at \( y = 0 \).

Would you like more details on function transformations or asymptote shifts? 

### Related Questions
1. How do you determine shifts of a function given an asymptote change?
2. What effect does a horizontal shift have on a function’s asymptotes?
3. How do vertical shifts affect the horizontal asymptote of a function?
4. How can you graphically identify horizontal and vertical shifts in a transformed function?
5. What other types of transformations could be applied to a base function?

### Tip
Remember, for a transformation \( y = f(x - h) + k \), \( h \) shifts the graph horizontally, while \( k \) shifts it vertically.

Let y = f(x) be the function whose graph is given in the upper-left in the figure. Find formulas, in terms of f(x), for the horizontal and vertical shifts of the graph of f(x) in parts (a)-(c) and determine the equation of each asymptote. Note only vertical and horizontal shifts have been applied to the graph of f (no reflections, compressions, or stretches), so your formulas should be of the form y = f(x − h) + k where h and k are constants you determine from each graph. (c) Using shifts of f(x), graph C can be written as: y = ? with an asymptote at y = ?

Explore how to find the formulas for horizontal and vertical shifts of a function, f(x), using graph transformations. Learn to determine the equation of each asymptote based on shifts in the graph. Suitable for high school students in Grades 10-12.

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