Final answer:
To create the new function g(x), the original absolute value function is transformed by reflecting it over the x-axis, stretching it vertically by a factor of 4, shifting it left by 1 unit, and shifting it up by 2 units, resulting in g(x) = -4| x + 1 | + 2.
Step-by-step explanation:
The student asked how to find the new equation g(x) for an absolute value function, f(x), after applying several transformations. To apply the transformations described (reflection over the x-axis, vertical stretch by 4, shift left 1 unit, and shift up 2 units), we can start with the standard absolute value function, f(x) = |x|. Reflection over the x-axis multiplies the function by -1, a vertical stretch by 4 multiplies it by 4, a shift left 1 unit replaces x with (x + 1), and a shift up by 2 units adds 2 to the entire function.
The transformed function g(x) would thus be:
g(x) = -4| x + 1 | + 2.
This represents the combined transformations acting on the original absolute value function.