1. The problem is to understand the graph of the function $$g(x) = (0.5)^{x + 3} - 4$$. 2. This is an exponential decay function because the base $$0.5$$ is between 0 and 1. 3. The general form of an exponential function is $$f(x) = a^{x + h} + k$$ where: - $$a$$ is the base, - $$h$$ shifts the graph horizontally, - $$k$$ shifts the graph vertically. 4. For $$g(x)$$, the base $$a = 0.5$$, the horizontal shift is $$-3$$ (because of $$x + 3$$), and the vertical shift is $$-4$$. 5. The horizontal shift means the graph is shifted 3 units to the left. 6. The vertical shift means the horizontal asymptote moves from $$y=0$$ to $$y=-4$$. 7. Since $$0 < 0.5 < 1$$, the graph is decreasing (exponential decay). 8. As $$x \to \infty$$, $$g(x) \to -4$$, so the graph approaches the line $$y = -4$$ from above. 9. As $$x \to -\infty$$, $$g(x) \to \infty$$, so the graph starts very high on the left. 10. This matches the description of the graph shown: starting high on the left, decreasing, and approaching $$y = -4$$ as $$x$$ increases. Final answer: The graph of $$g(x) = (0.5)^{x + 3} - 4$$ is an exponential decay curve shifted left by 3 units and down by 4 units, with horizontal asymptote $$y = -4$$.