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.