1. **Stating the problem:** We need to graph the function $$f(x) = \frac{x^2 + x - 2}{2x^2 - 2x - 3}$$ and analyze its key features. 2. **Formula and rules:** This is a rational function where the numerator and denominator are polynomials. Important rules: - The function is undefined where the denominator is zero. - Vertical asymptotes occur at values of $x$ that make the denominator zero but not the numerator. - Horizontal or oblique asymptotes depend on the degrees of numerator and denominator. 3. **Factor numerator and denominator:** $$x^2 + x - 2 = (x + 2)(x - 1)$$ $$2x^2 - 2x - 3 = 2x^2 - 3 - 2x = (2x + 1)(x - 3)$$ 4. **Simplify the function:** $$f(x) = \frac{(x + 2)(x - 1)}{(2x + 1)(x - 3)}$$ 5. **Find domain restrictions:** Denominator zero at: $$2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$$ $$x - 3 = 0 \Rightarrow x = 3$$ So, $x \neq -\frac{1}{2}, 3$. 6. **Find zeros of the function:** Numerator zero at: $$x + 2 = 0 \Rightarrow x = -2$$ $$x - 1 = 0 \Rightarrow x = 1$$ 7. **Horizontal asymptote:** Degree numerator = 2, degree denominator = 2. Leading coefficients ratio: $$\frac{1}{2}$$ So horizontal asymptote is: $$y = \frac{1}{2}$$ 8. **Summary of key points:** - Vertical asymptotes at $x = -\frac{1}{2}$ and $x = 3$ - Zeros at $x = -2$ and $x = 1$ - Horizontal asymptote at $y = \frac{1}{2}$ 9. **Graphing:** The function behaves like a rational function with these asymptotes and zeros.