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 opening upwards with its vertex at the origin \((0,0)\).
2. Recognize that the function \(g(x) = -|x|\) involves multiplying the output of \(f(x)\) by \(-1\), which affects the vertical direction of the graph.
3. Multiplying by \(-1\) reflects the graph of \(f(x) = |x|\) across the x-axis, turning all positive y-values into negative y-values and vice versa.
4. Therefore, the graph of \(g(x) = -|x|\) is a V-shaped graph opening downwards with its vertex still at the origin \((0,0)\).
5. In summary, to obtain the graph of \(g(x) = -|x|\) from \(f(x) = |x|\), reflect the entire graph of \(f(x)\) over the x-axis.

Key Concepts

Absolute Value Function

The absolute value function, f(x) = |x|, outputs the distance of x from zero, always producing non-negative values. Its graph is a V-shaped curve with the vertex at the origin (0,0), opening upwards. Understanding this base graph is essential for analyzing transformations applied to it.

Reflection Across the x-axis

Multiplying a function by -1 reflects its graph across the x-axis. For g(x) = -|x|, this means the V-shaped graph of |x| is flipped upside down, opening downward. This transformation changes all positive y-values to negative, altering the graph's orientation.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting a base graph to obtain a new graph. Recognizing how operations like negation affect the original function helps in sketching and understanding the new function's behavior quickly and accurately.