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

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1. Identify the transformation applied to the function $f(x)$. The function $g(x)=f(x-1)-1$ involves two transformations.
2. The expression $f(x-1)$ represents a horizontal shift. Specifically, it shifts the graph of $f(x)$ to the right by 1 unit.
3. The expression $-1$ outside the function represents a vertical shift. It shifts the graph of $f(x-1)$ downward by 1 unit.
4. To graph $g(x)$, start by taking the graph of $f(x)$ and shifting every point 1 unit to the right.
5. After shifting the graph to the right, move every point on the new graph 1 unit downward to complete the transformation.

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's graph.

Horizontal Shifts

Horizontal shifts occur when the input variable of a function is altered, affecting the graph's position along the x-axis. For g(x) = f(x-1), the 'x-1' indicates a shift 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 graph of g.

Vertical Shifts

Vertical shifts involve 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 a downward shift by 1 unit. This means that after applying the horizontal shift, each point on the graph of f(x) will be lowered by 1 unit, which is necessary for accurately representing g(x).