Graph each function.
$\text{ƒ}\left(x\right)=-\mid x\mid$

Verified step by step guidance

1. Understand the parent function: The function ƒ(x) = |x| represents the absolute value of x, which forms a V-shaped graph opening upwards with its vertex at the origin (0,0).
2. Identify the transformation: The given function is ƒ(x) = -|x|, which means the absolute value function is multiplied by -1. This reflects the graph of |x| across the x-axis.
3. Determine the vertex: Since there are no horizontal or vertical shifts, the vertex remains at the origin (0,0).
4. Plot key points: Choose values of x such as -2, -1, 0, 1, and 2. Calculate ƒ(x) = -|x| for each to get points like (-2, -2), (-1, -1), (0, 0), (1, -1), and (2, -2).
5. Draw the graph: Connect the plotted points with straight lines forming an upside-down V shape, opening downward, with the vertex at the origin.

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the distance of x from zero on the number line, always yielding a non-negative result. Its graph is a V-shaped curve with the vertex at the origin (0,0), opening upwards.

Reflection Across the x-axis

Multiplying a function by -1 reflects its graph across the x-axis. For ƒ(x) = -|x|, the standard V-shape of |x| is flipped upside down, creating an inverted V that opens downward.

Graphing Piecewise Functions

The absolute value function can be expressed as a piecewise function: |x| = x if x ≥ 0, and -x if x < 0. Understanding this helps in plotting points accurately and visualizing how the negative sign affects each piece.