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.