In Exercises 17-32, use the graph of y = f(x) to graph each function g.
g(x) = f(x-1)

Verified step by step guidance

1. Identify the transformation: g(x) = f(x-1) represents a horizontal shift of the graph of f(x) to the right by 1 unit.
2. Locate key points on the graph of f(x). For example, the vertex is at (0, -16), and the x-intercepts are at (-4, 0) and (4, 0).
3. Apply the horizontal shift to each key point. The vertex (0, -16) becomes (1, -16), and the x-intercepts (-4, 0) and (4, 0) become (-3, 0) and (5, 0), respectively.
4. Sketch the new graph of g(x) by plotting the shifted key points and maintaining the shape of the original graph.
5. Ensure the new graph is a parabola opening upwards, similar to the original graph, but shifted 1 unit to the right.

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function when it is altered by operations such as shifting, reflecting, or stretching. In this case, g(x) = f(x-1) represents a horizontal shift of the function f(x) to the right by 1 unit. Understanding how these transformations affect the graph is crucial for accurately sketching the new function.

Graph of a Parabola

The graph of a parabola is a U-shaped curve that can open upwards or downwards, depending on the leading coefficient of the quadratic function. In the provided graph, the function f(x) is a downward-opening parabola with its vertex at (0, -16) and x-intercepts at (-4, 0) and (4, 0). Recognizing the characteristics of parabolas helps in predicting how transformations will affect their shape and position.

Vertex and Intercepts

The vertex of a parabola is the highest or lowest point on the graph, while the x-intercepts are the points where the graph crosses the x-axis. For the function f(x), the vertex is at (0, -16), indicating it is the minimum point, and the x-intercepts at (-4, 0) and (4, 0) show where the function equals zero. Identifying these key points is essential for accurately graphing the transformed function g(x).