In Exercises 1-16, use the graph of y = f(x) to graph each function g.
g(x) = f(-x)+3

Key Concepts

Function Transformation

Function transformations involve changing the position or shape of a graph through various operations. In this case, the function g(x) = f(-x) + 3 represents a horizontal reflection of f(x) across the y-axis, followed by a vertical shift upwards by 3 units. Understanding these transformations is crucial for accurately graphing the new function based on the original.

Reflection Across the Y-Axis

Reflecting a function across the y-axis means that for every point (x, y) on the graph of f(x), there is a corresponding point (-x, y) on the graph of g(x). This transformation alters the x-coordinates of the function while keeping the y-coordinates the same, effectively flipping the graph horizontally. This concept is essential for visualizing how g(x) relates to f(x).

Vertical Shift

A vertical shift occurs when a function is moved up or down on the graph without changing its shape. In the function g(x) = f(-x) + 3, the '+3' indicates that the entire graph of f(-x) is shifted upwards by 3 units. This shift affects all y-values of the function, making it important to adjust the graph accordingly after applying the reflection.