The graph you provided appears to be the plot of a natural logarithmic function. Natural logarithmic functions have the general form:

$y = \ln(x)$

However, based on the graph and the marked point $\left(\frac{1}{e}, -3\right)$, the equation seems to have been vertically shifted and possibly scaled.

Observations:

1. The marked point $\left(\frac{1}{e}, -3\right)$ gives us a clue. The natural logarithm of $\frac{1}{e}$ is $-1$, so the function has been transformed.
2. The graph is shifted down by 2 units since $\ln\left(\frac{1}{e}\right) = -1$ for the regular natural logarithm, but here the $y$-value is $-3$. This indicates a vertical shift by 2 units down.
3. There also appears to be a vertical scaling, as the graph is stretched or compressed. Normally, $\ln\left(\frac{1}{e}\right) = -1$, but here the corresponding $y$-value is $-3$, suggesting a scaling factor.

Equation form:

This looks like a transformed logarithmic function of the form:

$y = a \ln(x) + b$

Using the point $\left(\frac{1}{e}, -3\right)$, we can substitute to solve for $a$ and $b$: $-3 = a \ln\left(\frac{1}{e}\right) + b$ Since $\ln\left(\frac{1}{e}\right) = -1$, this becomes: $-3 = -a + b$

Now, to confirm the vertical shift, observe that the graph appears to cross $y = 0$ at approximately $x = 1$, implying $b = 2$ (since $\ln(1) = 0$).

Thus, the equation becomes: $y = 2 \ln(x) - 2$

This seems to fit the overall behavior of the graph shown in your image.

Would you like further explanation or have any other questions?

Relative Questions:

1. How does shifting affect the graph of a logarithmic function?
2. What is the significance of the natural logarithm base $e$ in calculus?
3. How do you determine vertical scaling from a graph of a function?
4. Can you explain the properties of transformations applied to logarithmic functions?
5. What is the domain of the function $y = 2 \ln(x) - 2$?

Tip:

Always check the point where the logarithmic function equals zero, as it helps identify vertical shifts and other transformations in logarithmic graphs.