1. **State the problem:** We are given the function $$h(x) = -4|x - 5|$$ and need to identify the value of $$a$$ and the vertex of the graph. 2. **Formula and explanation:** The general form of an absolute value function is $$f(x) = a|x - h| + k$$ where: - $$a$$ controls the vertical stretch/compression and reflection. - $$(h,k)$$ is the vertex of the graph. 3. **Identify the value of $$a$$:** In $$h(x) = -4|x - 5|$$, the coefficient $$a = -4$$. - The negative sign means the graph is reflected over the x-axis (opens downward). - The absolute value causes a V-shaped graph. - The number 4 means the graph is vertically stretched by a factor of 4. 4. **Find the vertex:** The expression inside the absolute value is $$x - 5$$, so $$h = 5$$. - Since there is no $$+k$$ outside the absolute value, $$k = 0$$. - Therefore, the vertex is at the point $$(5,0)$$. 5. **Summary:** - The value of $$a$$ is $$-4$$. - The vertex is at $$(5,0)$$. This matches the description: a V-shaped graph, vertically stretched by 4, reflected over the x-axis, vertex at (5,0), opening downward.