1. The problem is to graph the function $$h(x) = -3|x + 3| - 2$$. 2. This is an absolute value function transformed by a vertical stretch/compression, reflection, and translation. 3. The general form of an absolute value function is $$f(x) = a|x - h| + k$$ where: - $$a$$ controls vertical stretch and reflection (if negative). - $$h$$ shifts the graph horizontally. - $$k$$ shifts the graph vertically. 4. For $$h(x) = -3|x + 3| - 2$$, rewrite $$x + 3$$ as $$x - (-3)$$, so: - $$a = -3$$ (vertical stretch by 3 and reflection over x-axis) - $$h = -3$$ (shift left by 3 units) - $$k = -2$$ (shift down by 2 units) 5. The vertex of the graph is at $$(-3, -2)$$. 6. The graph opens downward because $$a$$ is negative. 7. To plot points, choose values of $$x$$ around $$-3$$: - At $$x = -3$$, $$h(-3) = -3|0| - 2 = -2$$ (vertex) - At $$x = -2$$, $$h(-2) = -3|1| - 2 = -3 - 2 = -5$$ - At $$x = -4$$, $$h(-4) = -3| -1| - 2 = -3 - 2 = -5$$ - At $$x = 0$$, $$h(0) = -3|3| - 2 = -9 - 2 = -11$$ 8. The graph is a "V" shape reflected downward, vertex at $$(-3, -2)$$, and steepness controlled by 3. Final answer: The function $$h(x) = -3|x + 3| - 2$$ is an absolute value graph reflected over the x-axis, vertically stretched by 3, shifted left 3 units and down 2 units, with vertex at $$(-3, -2)$$.