In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = -f(x+2)

Verified step by step guidance

1. Identify the transformation components in the function g(x) = -f(x+2).
2. Recognize that f(x+2) represents a horizontal shift of the graph of y = f(x) to the left by 2 units.
3. Understand that the negative sign in front of f(x+2) indicates a reflection across the x-axis.
4. Combine the transformations: first shift the graph of y = f(x) left by 2 units, then reflect it across the x-axis.
5. Sketch the transformed graph based on these transformations to obtain the graph of g(x).

Key Concepts

Function Transformation

Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, reflecting, or stretching. In this case, the function g(x) = -f(x+2) involves both a horizontal shift and a vertical reflection of the original function f(x). Understanding these transformations is crucial for accurately graphing the new function.

Horizontal Shift

A horizontal shift occurs when a function is moved left or right on the Cartesian plane. For the function g(x) = -f(x+2), the term (x+2) indicates a shift of the graph of f(x) to the left by 2 units. This concept is essential for determining the new position of the graph after the transformation.

Vertical Reflection

Vertical reflection involves flipping the graph of a function over the x-axis. In the function g(x) = -f(x+2), the negative sign before f indicates that the graph of f(x) will be reflected vertically. This transformation changes the sign of the output values, which is important for accurately representing the new function's behavior.