Use the graph of y = f(x) to graph each function g.

g(x) = f(x) - 1

Verified step by step guidance

1. Step 1: Understand the transformation g(x) = f(x) - 1. This represents a vertical shift of the graph of f(x) downward by 1 unit. Each point on the graph of f(x) will have its y-coordinate decreased by 1.
2. Step 2: Identify key points on the graph of f(x). The given graph has three notable points: (-3, 0), (3, 0), and (0, -9). These points will be transformed according to the vertical shift.
3. Step 3: Apply the transformation to each key point. For the point (-3, 0), subtract 1 from the y-coordinate to get (-3, -1). For the point (3, 0), subtract 1 from the y-coordinate to get (3, -1). For the point (0, -9), subtract 1 from the y-coordinate to get (0, -10).
4. Step 4: Plot the transformed points on the graph. The new points are (-3, -1), (3, -1), and (0, -10). These points represent the graph of g(x).
5. Step 5: Draw the graph of g(x) by connecting the transformed points smoothly, maintaining the same shape as the original graph of f(x), but shifted downward 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 represents a vertical shift downward by 1 unit. Understanding how transformations affect the graph is crucial for accurately sketching the new function.

Graphing Quadratic Functions

Quadratic functions, typically in the form f(x) = ax^2 + bx + c, produce parabolic graphs. The given function f(x) has roots at (-3, 0) and (3, 0), indicating it opens upwards. Recognizing the shape and key features of quadratic graphs is essential for applying transformations correctly.

Identifying Key Points

Key points on a graph, such as intercepts and vertices, are critical for sketching functions accurately. In the provided graph, points like (0, -9) represent the vertex, while (-3, 0) and (3, 0) are x-intercepts. Identifying these points helps in understanding how the graph of g(x) will shift from f(x).