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

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.