Graph each function. ƒ(x) = -|x|

Verified step by step guidance

1. Identify the basic form of the function: The function given is \( f(x) = -|x| \), which is a transformation of the absolute value function \( |x| \).
2. Understand the transformation: The negative sign in front of \( |x| \) indicates a reflection over the x-axis. This means that instead of opening upwards like the standard absolute value function, the graph will open downwards.
3. Determine the vertex: The vertex of the absolute value function \( |x| \) is at the origin (0,0). Since there are no horizontal or vertical shifts in the function \( f(x) = -|x| \), the vertex remains at (0,0).
4. Sketch the graph: Start by plotting the vertex at (0,0). From the vertex, draw two lines that extend downwards at a 45-degree angle to the x-axis, forming a 'V' shape that opens downwards.
5. Label the axes and the graph: Ensure the x-axis and y-axis are clearly labeled, and label the graph as \( f(x) = -|x| \) to indicate the function being represented.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means that for any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. Understanding this function is crucial for graphing transformations, as it forms the basis for the shape of the graph.

Transformation of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. In the case of ƒ(x) = -|x|, the negative sign indicates a reflection over the x-axis. This concept is essential for predicting how the graph of a function will change based on modifications to its equation.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape and behavior of functions on a coordinate plane. For ƒ(x) = -|x|, the graph will form a 'V' shape that opens downward, with its vertex at the origin (0,0). Mastery of graphing techniques allows for accurate visual representation and analysis of functions.