To identify the correct graph for the function $f(x) = \log_2 x - 1$, let's analyze its properties:

1. Basic Shape of Logarithmic Function: The function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ and increases as $x$ increases. The graph generally passes through $(1, 0)$ for the function $\log_2 x$.

2. Vertical Shift: The subtraction of 1 in $f(x) = \log_2 x - 1$ results in a vertical downward shift by 1 unit. Therefore, instead of passing through $(1, 0)$, it will pass through $(1, -1)$.

3. Key Characteristics:

   • The graph should have a vertical asymptote at $x = 0$.
   • It should pass through $(1, -1)$ due to the downward shift.
   • It should be increasing for $x > 0$, showing an upward trend as $x$ moves to the right.

Looking at the four graphs:

• The top-left graph appears to match these characteristics: it has a vertical asymptote at $x = 0$, passes through $(1, -1)$, and shows an increasing trend as $x$ increases.

Thus, the correct graph is the top-left option.

Would you like further explanation, or have any questions about logarithmic transformations?

Here are five related questions to deepen your understanding:

1. How would the graph change if the function were $f(x) = \log_2(x + 1) - 1$?
2. What would happen to the graph if the base of the logarithm changed to a number less than 1, like $f(x) = \log_{0.5} x - 1$?
3. How does a horizontal shift affect the appearance of logarithmic functions?
4. What are the domain and range of the function $f(x) = \log_2 x - 1$?
5. How can you determine the asymptote of a logarithmic function just by looking at its equation?

Tip: When analyzing transformations, remember that shifts directly added or subtracted from $x$ affect the graph horizontally, while shifts added or subtracted outside of $x$ (like $-1$ here) affect it vertically.