In Exercises 1-16, use the graph of y = f(x) to graph each function g.
g(x) = f(x-1) - 2

Verified step by step guidance

1. Identify the transformation: g(x) = f(x-1) - 2 involves a horizontal shift to the right by 1 unit and a vertical shift down by 2 units.
2. Start with the original graph of y = f(x), which is a horizontal line segment from (1, -3) to (4, -3).
3. Apply the horizontal shift: move each point of the line segment 1 unit to the right, changing the points to (2, -3) and (5, -3).
4. Apply the vertical shift: move each point of the new line segment 2 units down, changing the points to (2, -5) and (5, -5).
5. Graph the new line segment for g(x) from (2, -5) to (5, -5) on the coordinate plane.

Key Concepts

Function Transformation

Function transformations involve changing the position or shape of a graph through various operations. In this case, g(x) = f(x-1) - 2 represents a horizontal shift to the right by 1 unit and a vertical shift downward by 2 units. 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 graph of f(x) is shifted to the right by 1 unit. This means that every point on the graph of f(x) will move right, which is essential for determining the new coordinates of g(x).

Vertical Shift

A vertical shift changes the position of a graph along the y-axis. In the function g(x) = f(x-1) - 2, the '-2' indicates that the entire graph of f(x) is moved down by 2 units. This transformation affects the y-coordinates of all points on the graph, which is necessary for accurately plotting g(x) after applying the horizontal shift.