In Exercises 17-32, 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 needed for g(x) = f(x-1) + 2, which involves a horizontal shift and a vertical shift.
2. Recognize that f(x-1) represents a horizontal shift of the graph of f(x) to the right by 1 unit.
3. Understand that adding 2 to f(x-1) results in a vertical shift of the graph upward by 2 units.
4. Apply the horizontal shift: move each point on the graph of f(x) to the right by 1 unit.
5. Apply the vertical shift: move each point on the graph of f(x-1) upward by 2 units to obtain the graph of g(x).

Key Concepts

Function Transformation

Function transformations involve altering the graph of a function through shifts, stretches, or reflections. In the case of g(x) = f(x-1) + 2, the graph of f(x) is shifted right by 1 unit and then moved up by 2 units. Understanding these transformations is crucial for accurately graphing the new function based on the original.

Horizontal Shifts

A horizontal shift occurs when the input of a function is adjusted by adding or subtracting a constant. For g(x) = f(x-1), the '-1' indicates a shift to the right. This means that every point on the graph of f(x) will move one unit to the right, affecting the x-coordinates of all points on the graph.

Vertical Shifts

Vertical shifts involve adding or subtracting a constant to the output of a function, which moves the graph up or down. In g(x) = f(x-1) + 2, the '+2' indicates that the entire graph of f(x) is raised by 2 units. This transformation affects the y-coordinates of all points on the graph, resulting in a new position for the graph of g.