The function f(x) = |x| transforms into g(x) = |x + 2| - 7 through a leftward shift by 2 units, a potential reflection, and a downward shift by 7 units, altering its position and shape.
To transform the function f(x) = |x| into g(x) = |x + 2| - 7, several key transformations occur in a specific order.
Translation Left:
The term x + 2 inside the absolute value function implies a horizontal shift to the left by 2 units, changing the position of the "peak" of the absolute value graph.
Reflection (if needed):
Since f(x) = |x| is already symmetric about the y-axis, no reflection is required. However, if the original function were f(x) = x^2, for example, a reflection would be necessary.
Vertical Shift Down:
The constant term -7 outside the absolute value affects the entire graph by vertically shifting it downward by 7 units.
Combining these transformations, the sequence is as follows:
g(x) = |x + 2| - 7
This transformation sequence ensures that the graph retains the V-shape characteristic of the absolute value function but shifts left, possibly reflects, and then shifts downward.