Question

What transformations are needed in order to obtain the function h(x) = |x + 2| - 5h(x)=∣x+2∣−5 from the parent function f(x) = |x|f(x)=∣x∣?

Answer 1

To get h(x) = |x + 2| - 5 from f(x) = |x|, a horizontal shift to the left by 2 units and a vertical shift downward by 5 units are required. These shifts correspond to the transformations inside and outside the absolute value function, respectively.

Step-by-step explanation:

To obtain the function h(x) = |x + 2| - 5 from the parent function f(x) = |x|, two transformations are required:

1. A horizontal shift: The term inside the absolute value changes from |x| to |x + 2|, which indicates a horizontal shift to the left by 2 units. In general, f(x - d) corresponds to a shift to the right by d units, while f(x + d) is a shift to the left by d units. Thus, x + 2 represents a leftward shift of the graph by 2 units.
2. A vertical shift: Subtracting 5 from |x + 2| to obtain |x + 2| - 5 signifies a downward shift of the entire graph by 5 units. This is because adding or subtracting a constant from the function affects the y-values directly, translating the graph vertically.