Describe how the graph of each function can be obtained from the graph of ƒ(x) = |x|. g(x) = -|x|

Verified step by step guidance

1. <Start with the basic graph of the function \( f(x) = |x| \), which is a V-shaped graph with its vertex at the origin (0,0).>
2. <The graph of \( f(x) = |x| \) is symmetric about the y-axis and opens upwards.>
3. <To obtain the graph of \( g(x) = -|x| \), reflect the graph of \( f(x) = |x| \) across the x-axis.>
4. <This reflection changes the direction of the V-shape, making it open downwards instead of upwards.>
5. <The vertex of the graph \( g(x) = -|x| \) remains at the origin (0,0), but the arms of the V now point downwards.>

Key Concepts

Absolute Value Function

The absolute value function, denoted as ƒ(x) = |x|, outputs the non-negative value of x. Its graph is a V-shape that opens upwards, with the vertex at the origin (0,0). This function is essential for understanding transformations, as it serves as the base graph from which other functions can be derived.

Vertical Reflection

A vertical reflection occurs when a graph is flipped over the x-axis. For the function g(x) = -|x|, the negative sign in front of the absolute value indicates that the graph of ƒ(x) = |x| is reflected downwards. This transformation changes the orientation of the graph, resulting in a V-shape that opens downwards.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. Understanding these transformations allows one to predict how the graph of a function will change based on modifications to its equation. In this case, the transformation from ƒ(x) = |x| to g(x) = -|x| illustrates a reflection, which is a fundamental concept in analyzing function behavior.