In Exercises 55–59, use the graph of to graph each function g.
g(x) = f(x + 2) + 3

Verified step by step guidance

1. Identify the original function f(x) from the graph. Here, f(x) is a V-shaped graph with its vertex at (0,0).
2. Understand the transformation g(x) = f(x + 2) + 3. This involves two transformations: a horizontal shift and a vertical shift.
3. First, apply the horizontal shift. The term (x + 2) indicates a shift to the left by 2 units. So, shift the entire graph of f(x) 2 units to the left.
4. Next, apply the vertical shift. The +3 outside the function indicates a shift upwards by 3 units. So, shift the entire graph (after the horizontal shift) 3 units up.
5. Plot the new graph g(x) by combining both transformations. The vertex of the new graph will be at (-2, 3), and the shape of the graph will remain the same as f(x).

Key Concepts

Function Transformation

Function transformations involve altering the graph of a function through shifts, stretches, or reflections. In this case, the function g(x) = f(x + 2) + 3 represents a horizontal shift to the left by 2 units and a vertical shift upward by 3 units. Understanding these transformations is crucial for accurately graphing the new function based on the original.

Vertex of a Function

The vertex of a function, particularly for quadratic functions like f(x), is the point where the graph changes direction. For the given function f(x), the vertex is at (0, 0). When transforming the function to g(x), the vertex will also shift according to the transformations applied, which is essential for determining the new position of the graph.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of the function to create an accurate representation. For g(x), after applying the transformations to f(x), one can plot the new vertex and additional points to visualize the complete graph. Familiarity with these techniques helps in effectively translating the original graph to the transformed graph.