1. **Problem statement:** We are given the function $$f(x) = \frac{5x - 1}{x^2 - 3x + 2}$$ and need to analyze it. 2. **Step 1: Factor the denominator.** The denominator is a quadratic expression: $$x^2 - 3x + 2$$ We look for two numbers that multiply to 2 and add to -3, which are -1 and -2. So, $$x^2 - 3x + 2 = (x - 1)(x - 2)$$ 3. **Step 2: Identify domain restrictions.** The function is undefined where the denominator is zero: $$x - 1 = 0 \Rightarrow x = 1$$ $$x - 2 = 0 \Rightarrow x = 2$$ So, the domain is all real numbers except $x = 1$ and $x = 2$. 4. **Step 3: Simplify the function if possible.** The numerator is $5x - 1$, which does not share factors with the denominator, so no simplification is possible. 5. **Step 4: Find vertical asymptotes.** Vertical asymptotes occur where the denominator is zero and the numerator is not zero: At $x=1$, numerator $= 5(1) - 1 = 4 \neq 0$, so vertical asymptote at $x=1$. At $x=2$, numerator $= 5(2) - 1 = 9 \neq 0$, so vertical asymptote at $x=2$. 6. **Step 5: Find horizontal asymptote.** Since degree of numerator (1) is less than degree of denominator (2), horizontal asymptote is $y=0$. 7. **Step 6: Find x-intercept(s).** Set numerator equal to zero: $$5x - 1 = 0 \Rightarrow x = \frac{1}{5}$$ So, x-intercept at $\left(\frac{1}{5}, 0\right)$. 8. **Step 7: Find y-intercept.** Evaluate $f(0)$: $$f(0) = \frac{5(0) - 1}{0^2 - 3(0) + 2} = \frac{-1}{2} = -\frac{1}{2}$$ So, y-intercept at $(0, -\frac{1}{2})$. **Final answer:** - Domain: $x \neq 1, 2$ - Vertical asymptotes: $x=1$ and $x=2$ - Horizontal asymptote: $y=0$ - x-intercept: $\left(\frac{1}{5}, 0\right)$ - y-intercept: $(0, -\frac{1}{2})$