Final answer:
The absolute value function that describes a vertex at (3,-5), opening down, stretched by a factor of 2 is f(x) = -2|x - 3| - 5. We shift the basic absolute value function to the right and downwards, reflect it over the x-axis, and stretch it vertically to obtain the desired transformation.
Step-by-step explanation:
To write an absolute value function that represents the described transformation, we need to apply our knowledge of function transformations. The basic absolute value function is f(x) = |x|. To translate the vertex to the point (3,-5), we must shift the function 3 units to the right and 5 units down. This is achieved by replacing x with x - 3 and subtracting 5 from the entire function, yielding f(x) = |x - 3| - 5.
Since the function is opening down and is stretched by a factor of 2, we need to introduce a negative sign to represent the reflection over the x-axis, and multiply the absolute value expression by 2 for the vertical stretch. This results in the final function f(x) = -2|x - 3| - 5.