Final answer:
The best equation to represent the flight of the geese in the form y = a |(x - h)| + k is B. y = 2|x - 5| + 3, where the point of reference (5, 3) is used within the absolute value function to represent the lead goose's east and north coordinates from the bird watcher.
Step-by-step explanation:
The student is asking for the equation that best represents the flight of two geese in the form of y = a |(x - h)| + k, where the lead goose's position is 5 miles east and 3 miles north and the second goose's position is 1 mile east and 2 miles north. Looking at the provided options, the equation needs to reflect the principle of absolute value to represent the flight path of the geese relative to the bird watcher at the origin.
Given that the lead goose is 5 miles east, this means that it's 5 units on the x-axis, and since it's flying north this relates to the y-coordinate. The 'h' in the equation will be the x-coordinate of the goose’s position, and 'k' will be the y-coordinate. The rise over run will give us the 'a' or slope. For the lead goose, the slope of the flight path is 3/5. However, because slope doesn't apply directly in an absolute value equation that's not in slope-intercept form, it's more important to focus on the structure of the function, which would center on the point (5, 3) for the lead goose. As such, option B, y = 2|x - 5| + 3, is the one that reflects this position correctly, where 'h' and 'k' have been replaced with 5 and 3, respectively.