In Exercises 45-52, use the graph of y = f(x) to graph each function g. g(x) = -f(x-1) + 1

Verified step by step guidance

1. Step 1: Understand the transformation involved in g(x) = -f(x-1) + 1.
2. Step 2: Recognize that f(x-1) represents a horizontal shift of the graph of f(x) to the right by 1 unit.
3. Step 3: Apply the negative sign in front of f(x-1), which reflects the graph across the x-axis.
4. Step 4: Finally, add 1 to the entire function, which shifts the graph vertically upward by 1 unit.
5. Step 5: Combine all transformations to graph g(x) by shifting f(x) right by 1 unit, reflecting it across the x-axis, and then shifting it up by 1 unit.

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, a reflection, and a vertical shift, which alters the original graph of f(x) to create g(x). Understanding these transformations is crucial for accurately graphing the new function.

Horizontal Shift

A horizontal shift occurs when the input of a function is adjusted, affecting the graph's position along the x-axis. In g(x) = -f(x-1) + 1, the term (x-1) indicates a shift to the right by 1 unit. This means that every point on the graph of f(x) will move rightward, which is essential for determining the new graph of g(x).

Vertical Shift and Reflection

A vertical shift occurs when a constant is added or subtracted from a function, moving the graph up or down. In g(x) = -f(x-1) + 1, the '+1' shifts the graph of -f(x-1) upward by 1 unit. Additionally, the negative sign before f indicates a reflection over the x-axis, flipping the graph upside down. Both transformations are vital for accurately sketching the graph of g.