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

Verified step by step guidance

1. Identify the transformation: g(x) = f(x+1) represents a horizontal shift to the left by 1 unit.
2. Locate key points on the graph of y = f(x). For example, the endpoints of the line segment are at (1, 2) and (3, 2).
3. Apply the horizontal shift to each key point by subtracting 1 from the x-coordinate. The new points will be (0, 2) and (2, 2).
4. Draw the new graph of g(x) by connecting the transformed points with a line segment, maintaining the same slope and length as the original segment.
5. Ensure the new graph is parallel to the original graph and shifted 1 unit to the left.

Key Concepts

Function Transformation

Function transformation refers to the process of altering the graph of a function through shifts, stretches, or reflections. In this case, the transformation involves a horizontal shift, which affects the x-coordinates of the function's graph. Understanding how to apply these transformations is crucial for accurately graphing the new function g(x) based on f(x).

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 + 1), the graph of f(x) is shifted to the left by 1 unit. This shift changes the x-values of all points on the graph, which is essential to visualize the new function correctly.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to represent the relationship between the input (x) and output (y) values. To graph g(x) = f(x + 1), one must first understand the original graph of f(x) and then apply the horizontal shift to each point. This skill is fundamental in algebra for visualizing and interpreting functions.