The function you're asked to graph is $f(x) = 5 \ln(x)$, which is a scaled version of the natural logarithmic function. Let's break down the key features of this graph:

1. Domain: The natural logarithmic function $\ln(x)$ is only defined for $x > 0$, so the graph will not exist for $x \leq 0$.

2. Intercepts:

   • The function will cross the x-axis at $x = 1$ because $\ln(1) = 0$, and scaling by 5 keeps the value at 0.
   • The y-axis is not in the domain of the function, but as $x$ approaches 0 from the right, the function approaches negative infinity.

3. Behavior:

   • As $x \to 0^+$, $\ln(x) \to -\infty$, so the graph will steeply drop towards negative infinity as it approaches $x = 0$.
   • As $x \to \infty$, the graph will rise at a decreasing rate since the logarithmic function grows slowly for large values of $x$.

4. Key Points:

   • At $x = 1$, the function crosses the x-axis: $f(1) = 5 \ln(1) = 0$.
   • The function increases as $x$ gets larger, but at a decreasing rate.

Now, let's examine the options based on these features:

• Option A describes a graph that rises steeply in quadrant 4, crosses the x-axis at 1, and rises at a decreasing rate with points approximately at (2, 1.4) and (3, 2.2). This matches the behavior of $f(x) = 5 \ln(x)$, so this is likely the correct choice.

• Option B describes points at (2, 3.5) and (3, 5.5), which is not consistent with the expected behavior of $f(x) = 5 \ln(x)$ as it grows slower than these values suggest.

• Option C describes points at (2, 2.8) and (3, 4.4), which is still too large for $f(x) = 5 \ln(x)$.

Therefore, Option A is the correct graph.

Would you like more details or clarifications on how the graph behaves? Here are some related questions for further exploration:

1. How do transformations affect the graph of a logarithmic function?
2. What is the general shape of the graph of $y = \ln(x)$?
3. How can you determine the horizontal asymptote of a logarithmic function?
4. How does the natural logarithmic function compare to other logarithmic functions, such as $\log(x)$?
5. How would you graph $f(x) = \ln(x)$ without a calculator?

Tip: When graphing logarithmic functions, always remember that the function is undefined for $x \leq 0$, and check for vertical asymptotes near $x = 0$.