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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. (a) Using shifts of 𝑓(𝑥), graph A can be written as: y=? with an asymptote at y=?

From the information provided and the images, we’re given a function \( y = f(x) \) and need to determine a formula for a shifted version of \( f(x) \) in the form \( y = f(x - h) + k \). We also need to identify the equation of the horizontal asymptote for the graph with the shift.

Let’s examine the graphs step by step.

1. **Original Graph Analysis**: The first graph represents \( f(x) \) with an asymptote at \( y = 0 \) (the x-axis). The function appears to start from a point just above \( x = 0 \) and increases toward a horizontal asymptote at \( y = 5 \) as \( x \to \infty \). 

2. **Shifted Graph Analysis**: The second graph shows a similar shape, suggesting it's a shifted version of \( f(x) \), but now the horizontal asymptote is at \( y = 5 \). This indicates a vertical shift upward by 5 units.

### Part (a): Determine the Shifted Function and Asymptote
The shifted graph can be written as:
\[
y = f(x) + 5
\]
This vertical shift moves the asymptote from \( y = 0 \) to \( y = 5 \).

### Final Answer for Part (a):
- The shifted function can be written as:
  \[
  y = f(x) + 5
  \]
- The equation of the horizontal asymptote is:
  \[
  y = 5
  \]

Would you like more details or further explanation?

### Follow-Up Questions:
1. How would the formula change if there was a horizontal shift instead?
2. What would the new asymptote be if we applied a vertical shift downward by 3 units?
3. How does a vertical shift affect the y-intercept of the graph?
4. Can we determine the exact form of \( f(x) \) from the given graph, or only shifts?
5. What effect would a reflection over the x-axis have on the asymptote?

#### Tip:
In problems with shifts, observe changes in asymptotes to identify vertical shifts quickly.

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. (a) Using shifts of f(x), graph A can be written as: y=? with an asymptote at y=?

Explore how to determine formulas for horizontal and vertical shifts of a function y = f(x), and how to identify asymptotes. This problem focuses on function transformations without reflections or stretches, suitable for high school math.

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