Graph the function, not by plotting points, but by starting from the graphs in the figures below. \[ f(x)=\log _{3}(x-3) \]

We begin with the basic logarithmic function \[ g(x)=\log_{3}(x) \] which has the following characteristics: - **Domain:** \( x>0 \) - **Vertical asymptote:** \( x=0 \) - **x-intercept:** \( (1,0) \) The given function is \[ f(x)=\log_{3}(x-3). \] Notice that the argument of the logarithm, \( x-3 \), means that we have a horizontal translation of the graph of \( g(x) \). **Step 1. Horizontal Shift** The transformation \( x \to x - 3 \) translates the basic graph to the right by 3 units. This means: - The vertical asymptote shifts from \( x=0 \) to \( x=3 \). - The x-intercept shifts from \( (1, 0) \) to \( (1+3, 0)=(4, 0) \). **Step 2. Domain and Asymptote** After the shift, the domain of \( f(x) \) becomes: \[ x-3>0 \quad \Longrightarrow \quad x>3. \] Thus, the vertical asymptote is now at \( x=3 \). **Step 3. Sketching the Graph Using the Transformed Features** - Draw the vertical line \( x=3 \) as the asymptote. - Mark the x-intercept at \( (4,0) \). - The graph will pass through the point obtained by shifting another key point. For example, in the graph of \( g(x) \), when \( x = 3 \) we have \( g(3)=\log_3 3=1 \). Shifting right by 3 gives the point \( (3+3, 1)=(6,1) \). Thus, the important characteristics of \( f(x)=\log_3 (x-3) \) are: - **Vertical asymptote:** \( x=3 \) - **x-intercept:** \( (4, 0) \) - **A known point:** \( (6, 1) \) - **Domain:** \( x>3 \) Using these transformed features from the original graph of \( \log_3(x) \), you can sketch the graph of \( f(x)=\log_3(x-3) \) without plotting individual points.