1. **State the problem:** We are given the function $f(x) = \frac{2x - 3}{-x + 3}$ and need to analyze its graph, including asymptotes and points. 2. **Identify vertical asymptote:** The vertical asymptote occurs where the denominator is zero. $$-x + 3 = 0 \implies x = 3$$ So, there is a vertical asymptote at $x = 3$. 3. **Identify horizontal asymptote:** For rational functions where numerator and denominator are degree 1, the horizontal asymptote is the ratio of leading coefficients. Leading coefficient numerator: 2 Leading coefficient denominator: -1 $$y = \frac{2}{-1} = -2$$ So, horizontal asymptote is $y = -2$. 4. **Plot points on each side of the vertical asymptote:** - For $x < 3$, choose $x = 0$ and $x = 2$: $$f(0) = \frac{2(0) - 3}{-0 + 3} = \frac{-3}{3} = -1$$ $$f(2) = \frac{2(2) - 3}{-2 + 3} = \frac{4 - 3}{1} = 1$$ - For $x > 3$, choose $x = 4$ and $x = 5$: $$f(4) = \frac{2(4) - 3}{-4 + 3} = \frac{8 - 3}{-1} = \frac{5}{-1} = -5$$ $$f(5) = \frac{2(5) - 3}{-5 + 3} = \frac{10 - 3}{-2} = \frac{7}{-2} = -3.5$$ 5. **Summary:** - Vertical asymptote at $x = 3$ - Horizontal asymptote at $y = -2$ - Points on left side: $(0, -1)$ and $(2, 1)$ - Points on right side: $(4, -5)$ and $(5, -3.5)$ This information helps sketch the graph showing the behavior near asymptotes and points.