Final answer:
The parent function is f(x) = |x|, and after analyzing the transformations for each option, the corrected transformed function is f(x) = 2|x - 3| - 5. This function represents a vertical stretch by a factor of 2, a horizontal shift to the right by 3 units, and a vertical shift down by 5 units.
Step-by-step explanation:
The parent function given is f(x) = |x|, which is the absolute value function. We are tasked with identifying the transformed function and describing the transformations that occur to create it. Let's look at the options and apply our knowledge of function transformations:
- Vertical Stretch or Compression: If we have a coefficient a before the absolute value (such as 2 in 2|x|), it means the graph is either stretched (if |a| > 1) or compressed (if |a| < 1) vertically by a factor of a.
- Reflections: If there's a negative sign before the absolute value (as in -|x|), the graph is reflected over the x-axis.
- Horizontal Shifts: If there's a subtraction or addition inside the absolute value (as in |x - h| or |x + h|), the graph is shifted horizontally by h units to the right (if -h) or to the left (if +h).
- Vertical Shifts: If a number is added or subtracted outside the absolute value (as in |x| + k), the graph is shifted upwards (if +k) or downwards (if -k) by k units.
Considering these transformations, the corrected transformed function is B. Parent Function: f(x) = |x|; Transformed Function: f(x) = 2|x - 3| - 5, which means the graph of f(x) = |x| is stretched vertically by a factor of 2, shifted to the right by 3 units, and then shifted downward by 5 units.