Given the following functions f(x) and g(x), describe the changes (horizontal shifts, vertical shifts, etc.) to the graph of g(x) with respect to the graph of f(x). *
f(x) = |x| g(x) = -|x+3| - 2
|x+3| ... shift left 3
-|x+3| ... reflect over x-axis
-|x+3|-2 ... shift down 2
To describe the changes between the graphs of f(x) = |x| and g(x) = -|x+3| - 2, we can break it down into the following transformations:
1. Horizontal Shift: In both functions, we have an expression in the form of |x + a|. The value of "a" represents a horizontal shift. In function f(x), there is no horizontal shift as a = 0. However, in g(x), a = -3, indicating a horizontal shift of 3 units to the left.
2. Vertical Shift: Both functions have a vertical shift represented by a constant value added to the main expression. In f(x), there is no vertical shift as the constant is zero. But in g(x), we have -2 added at the end, causing a vertical shift two units downward.
3. Reflection: In function f(x), the absolute value |x| results in a V-shaped graph that is not reflected. However, in g(x), the negative sign before the absolute value -|x+3| reflects the graph across the x-axis, making it an upside-down V-shaped graph.
To summarize, the changes between the graphs of f(x) and g(x) are:
- A horizontal shift of 3 units to the left.
- A vertical shift of 2 units downward.
- A reflection across the x-axis.