Question: Construct a function with the following conditions. • vertical asymptote at $x = 3$ • slant or oblique asymptote at $y = 3x + 7$ ○ $y = \frac{3x + 8}{x + 3}$ ○ $y = \frac{3x + 8}{x - 3}$ ○ $y = \frac{3x^2 - 2x - 20}{x + 3}$ ○ $y = \frac{3x^2 - 2x - 20}{x - 3}$ Graph shape: multiple-choice list on the left side of the page, with the problem statement at the top-left; the correct-function candidates are centered vertically in the middle-left to lower-left area, so position_hint = top-left.

1. **State the problem:** We need to find which function has a vertical asymptote at $x=3$ and a slant (oblique) asymptote at $y=3x+7$. 2. **Recall vertical asymptote rule:** Vertical asymptotes occur where the denominator is zero and the numerator is not zero at that point. 3. **Recall slant asymptote rule:** A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. The slant asymptote is found by performing polynomial division of numerator by denominator. 4. **Check each candidate function:** - For $y=\frac{3x+8}{x+3}$: - Vertical asymptote at $x=-3$, not $3$. - Degree numerator = 1, denominator = 1, so no slant asymptote. - For $y=\frac{3x+8}{x-3}$: - Vertical asymptote at $x=3$ (denominator zero). - Degree numerator = 1, denominator = 1, so no slant asymptote. - For $y=\frac{3x^2 - 2x - 20}{x+3}$: - Vertical asymptote at $x=-3$, not $3$. - Degree numerator = 2, denominator = 1, so slant asymptote possible. - Perform division: $$\frac{3x^2 - 2x - 20}{x+3} = 3x - 11 + \frac{13}{x+3}$$ - Slant asymptote is $y=3x - 11$, not $3x + 7$. - For $y=\frac{3x^2 - 2x - 20}{x-3}$: - Vertical asymptote at $x=3$. - Degree numerator = 2, denominator = 1, so slant asymptote possible. - Perform division: $$\frac{3x^2 - 2x - 20}{x-3} = 3x + 7 + \frac{1}{x-3}$$ - Slant asymptote is $y=3x + 7$, which matches the condition. 5. **Conclusion:** The function $$y=\frac{3x^2 - 2x - 20}{x-3}$$ has the vertical asymptote at $x=3$ and slant asymptote $y=3x+7$. **Final answer:** $$y=\frac{3x^2 - 2x - 20}{x-3}$$