The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.
(d) y = |ƒ(x)|

Verified step by step guidance

1. Identify the points where the function f(x) is negative. In this graph, f(x) is negative between x = 0 and x = 16.
2. For the portion of the graph where f(x) is negative, reflect these points across the x-axis to make them positive. This means that the point (8, -4) will become (8, 4).
3. Keep the portions of the graph where f(x) is already positive unchanged. This includes the segments from x = -8 to x = 0 and the point (0, 0).
4. Sketch the new graph by combining the unchanged positive portions with the reflected negative portions. The new graph will have a segment from (0, 0) to (8, 4) and then from (8, 4) to (16, 0).
5. Ensure the final graph represents y = |f(x)|, which means all y-values are non-negative.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |ƒ(x)|, transforms any negative output of the function ƒ into a positive value. This means that for any x where ƒ(x) is negative, the graph of |ƒ(x)| will reflect that portion of the graph above the x-axis, effectively making all y-values non-negative.

Graph Transformation

Graph transformation involves modifying the original graph of a function to create a new graph. In this case, applying the absolute value to the function ƒ results in a vertical reflection of the parts of the graph that lie below the x-axis, while the parts above remain unchanged. Understanding this concept is crucial for accurately sketching the new graph.

Critical Points and Intercepts

Critical points are specific points on the graph where the function changes behavior, such as local maxima, minima, or intercepts. For the function ƒ, the x-intercepts (where y=0) and the maximum point at (-4, 8) are essential for sketching |ƒ(x)|, as they determine where the graph touches or crosses the x-axis and where it reaches its highest values.