Use the graph of y = f(x) to graph each function g. g(x) = f(x-1) – 1

Verified step by step guidance

1. Understand the given problem: You are tasked with graphing the function g(x) = f(x-1) - 1 based on the graph of y = f(x). This involves applying transformations to the graph of f(x).
2. Identify the transformations: The term (x-1) inside the function indicates a horizontal shift to the right by 1 unit, and the term -1 outside the function indicates a vertical shift downward by 1 unit.
3. Apply the horizontal shift: Take each point (a, b) on the graph of y = f(x) and shift it to the right by 1 unit. This means the new x-coordinate will be a+1, while the y-coordinate remains b.
4. Apply the vertical shift: After applying the horizontal shift, take each new point and shift it downward by 1 unit. This means the new y-coordinate will be b-1, while the x-coordinate remains unchanged.
5. Plot the transformed points: Using the new coordinates obtained after both transformations, plot the points on the graph to create the graph of g(x). Ensure the shape of the graph remains consistent with the original graph of f(x).

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. In this case, g(x) = f(x-1) – 1 involves a horizontal shift to the right by 1 unit and a vertical shift downward by 1 unit. Understanding these transformations is crucial for accurately graphing the new function based on the original function f(x).

Horizontal Shift

A horizontal shift occurs when the input of a function is altered, affecting the graph's position along the x-axis. For g(x) = f(x-1), the 'x-1' indicates that the graph of f(x) is moved 1 unit to the right. This shift does not change the shape of the graph but repositions it, which is essential for correctly plotting g(x).

Vertical Shift

A vertical shift involves moving the graph of a function up or down along the y-axis. In the function g(x) = f(x-1) – 1, the '–1' indicates that the entire graph of f(x) is shifted down by 1 unit. This transformation is important for understanding how the output values of the function change relative to the original function.