Graph the rational function. $$ f(x)=\frac{3 x-3}{-x+2} $$ Start by drawing the vertical and horizontal asymptotes. Ther

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1. Find the vertical asymptote by setting the denominator equal to zero:
   $$
   -x+2=0 \quad \Rightarrow \quad x=2.
   $$
   So, there is a vertical asymptote at $$x=2$$.

2. Determine the horizontal asymptote. Since the degrees of the numerator and denominator are both 1, the horizontal asymptote is the ratio of the leading coefficients. The numerator’s leading coefficient is 3 and the denominator’s is -1:
   $$
   y=\frac{3}{-1}=-3.
   $$
   Thus, the horizontal asymptote is $$y=-3$$.

3. It is helpful to rewrite the function for further analysis. Start with
   $$
   f(x)=\frac{3x-3}{-x+2}.
   $$
   Factor the numerator and factor out $$-1$$ from the denominator:
   $$
   f(x)=\frac{3(x-1)}{-1(x-2)}=-\frac{3(x-1)}{x-2}.
   $$
   Next, express the fraction $$\frac{x-1}{x-2}$$ in a form that shows the transformation relative to the asymptotes:
   $$
   \frac{x-1}{x-2}=\frac{(x-2)+1}{x-2}=1+\frac{1}{x-2}.
   $$
   Therefore,
   $$
   f(x)=-3\left(1+\frac{1}{x-2}\right)=-3-\frac{3}{x-2}.
   $$

4. Identify additional key points of the graph:
   - **$$x$$-intercept:** Set $$f(x)=0$$ by solving the numerator of the original function:
     $$
     3x-3=0 \quad \Rightarrow \quad x=1.
     $$
     So, the $$x$$-intercept is $$(1,0)$$.
   - **$$y$$-intercept:** Set $$x=0$$:
     $$
     f(0)=\frac{3(0)-3}{-0+2}=\frac{-3}{2}=-\frac{3}{2}.
     $$
     Thus, the $$y$$-intercept is $$\left(0,-\frac{3}{2}\right)$$.

5. Graphing the function involves drawing the asymptotes and plotting the identified points:
   - Draw a vertical dashed line at $$x=2$$ (vertical asymptote).
   - Draw a horizontal dashed line at $$y=-3$$ (horizontal asymptote).
   - Plot the $$x$$-intercept at $$(1,0)$$ and the $$y$$-intercept at $$\left(0,-\frac{3}{2}\right)$$.
   - Sketch the curve using the factored form
     $$
     f(x)=-3-\frac{3}{x-2},
     $$
     keeping in mind the behavior near the asymptotes.

These steps provide the necessary information to begin graphing the function $$ f(x)=\frac{3 x-3}{-x+2} $$.
To graph the function $$ f(x) = \frac{3x - 3}{-x + 2} $$, follow these steps: 1. **Vertical Asymptote**: Set the denominator equal to zero to find $$ x = 2 $$. 2. **Horizontal Asymptote**: Since the degrees of numerator and denominator are the same, the horizontal asymptote is $$ y = -3 $$. 3. **Intercepts**: - $$ x $$-intercept at $$ x = 1 $$. - $$ y $$-intercept at $$ y = -\frac{3}{2} $$. 4. **Graph**: - Draw vertical and horizontal asymptotes at $$ x = 2 $$ and $$ y = -3 $$. - Plot the intercepts. - Sketch the curve using the factored form $$ f(x) = -3 - \frac{3}{x - 2} $$. This will help in accurately graphing the function.

Graph the rational function. \[ f(x)=\frac{3 x-3}{-x+2} \] Start by drawing the vertical and horizontal asymptotes. Ther

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