The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(a) y = ƒ(x) +3

Verified step by step guidance

1. Identify the original function ƒ(x) from the graph, noting key points such as (-8, 0), (-4, 8), (0, 0), (8, -4), and (16, 0).
2. Understand that the transformation y = ƒ(x) + 3 means you add 3 to the output (y-values) of the original function for every x.
3. Apply this vertical shift by adding 3 to the y-coordinate of each key point: for example, (-8, 0) becomes (-8, 0 + 3), (-4, 8) becomes (-4, 8 + 3), and so on.
4. Plot the new points on the graph using the adjusted y-values: (-8, 3), (-4, 11), (0, 3), (8, -1), and (16, 3).
5. Sketch the new graph by connecting these points smoothly, maintaining the original shape but shifted upward by 3 units.

Key Concepts

Function Transformation

Function transformation involves changing the graph of a function by shifting, stretching, compressing, or reflecting it. In this problem, adding a constant to the function, such as y = ƒ(x) + 3, shifts the graph vertically upward by 3 units without altering its shape.

Vertical Shift

A vertical shift moves the entire graph of a function up or down along the y-axis. For y = ƒ(x) + 3, every point (x, y) on the original graph moves to (x, y + 3), increasing the y-coordinate by 3. This shift does not affect the x-values or the shape of the graph.

Graph Interpretation

Interpreting a graph requires understanding key points and their coordinates. The given graph shows specific points of ƒ(x), which helps in sketching the transformed graph by applying the vertical shift to each point, ensuring accuracy in the new graph's shape and position.