1. Problem: Sketch the function $y = -2 \log_3 (x + 4)$ and state its domain, range, asymptotes, and intercepts. 2. Domain: The argument of the logarithm must be positive: $$x + 4 > 0 \implies x > -4$$ So, the domain is $(-4, \infty)$. 3. Range: Since the logarithm function can take any real value and is scaled by $-2$, the range remains all real numbers: $$\text{Range} = (-\infty, \infty)$$ 4. Vertical asymptote: The logarithm has a vertical asymptote where its argument is zero: $$x + 4 = 0 \implies x = -4$$ 5. Intercepts: - To find the $y$-intercept, set $x=0$: $$y = -2 \log_3 (0 + 4) = -2 \log_3 4$$ - To find the $x$-intercept, set $y=0$: $$0 = -2 \log_3 (x + 4) \implies \log_3 (x + 4) = 0$$ $$x + 4 = 3^0 = 1 \implies x = -3$$ 6. Sketch description: - The graph has a vertical asymptote at $x = -4$. - It passes through $(-3, 0)$ (the $x$-intercept). - It passes through $(0, -2 \log_3 4)$ (the $y$-intercept). - The graph is reflected vertically (due to the negative sign) and stretched by a factor of 2.