1. **State the problem:** We have a rational function which is a quotient of linear functions. It has a vertical asymptote at $x=-4$, a horizontal asymptote at $y=3$, and an $x$-intercept at $(8,0)$. We need to find the $y$-intercept. 2. **Set up the function:** Since the function is a quotient of linear functions, write it as $$f(x) = \frac{ax + b}{cx + d}$$ where $a,b,c,d$ are constants. 3. **Use vertical asymptote:** The vertical asymptote at $x=-4$ means the denominator is zero there: $$c(-4) + d = 0 \implies -4c + d = 0 \implies d = 4c$$ 4. **Use horizontal asymptote:** The horizontal asymptote $y=3$ means the ratio of leading coefficients is 3: $$\lim_{x \to \pm \infty} \frac{ax + b}{cx + d} = \frac{a}{c} = 3 \implies a = 3c$$ 5. **Use $x$-intercept:** The $x$-intercept at $(8,0)$ means: $$f(8) = 0 = \frac{a(8) + b}{c(8) + d} \implies a(8) + b = 0 \implies 8a + b = 0$$ 6. **Substitute $a=3c$ into $8a + b=0$:** $$8(3c) + b = 0 \implies 24c + b = 0 \implies b = -24c$$ 7. **Rewrite function with $a,b,d$ in terms of $c$:** $$f(x) = \frac{3cx - 24c}{cx + 4c} = \frac{c(3x - 24)}{c(x + 4)}$$ Cancel $c$ (assuming $c \neq 0$): $$f(x) = \frac{3x - 24}{x + 4}$$ 8. **Find $y$-intercept:** The $y$-intercept is $f(0)$: $$f(0) = \frac{3(0) - 24}{0 + 4} = \frac{-24}{4} = -6$$ **Final answer:** The $y$-intercept is $-6$.