Graph each function. ƒ(x) = |x| -3

Verified step by step guidance

1. Identify the basic function and transformations: The function ƒ(x) = |x| - 3 is a transformation of the basic absolute value function ƒ(x) = |x|. The transformation involves a vertical shift.
2. Understand the vertical shift: The '-3' in the function indicates a vertical shift downward by 3 units. This means that every point on the basic absolute value graph, ƒ(x) = |x|, will be moved down 3 units.
3. Plot the basic absolute value function: Start by plotting the basic graph of ƒ(x) = |x|. This graph has a vertex at (0,0) and forms a 'V' shape, with the left arm descending and the right arm ascending symmetrically from the vertex.
4. Apply the vertical shift: From the basic graph of ƒ(x) = |x|, shift every point down by 3 units. This will move the vertex from (0,0) to (0,-3).
5. Draw the transformed graph: After shifting the graph down, redraw the 'V' shaped graph with the new vertex at (0,-3). The arms of the 'V' will still have the same slope as the original absolute value function.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial for graphing ƒ(x) = |x| - 3, as it influences the overall shape and position of the graph.

Vertical Shifts

Vertical shifts occur when a function is adjusted up or down by adding or subtracting a constant from the function's output. In the function ƒ(x) = |x| - 3, the '-3' indicates a downward shift of the entire graph by three units. This concept is essential for accurately positioning the graph of the function relative to the standard absolute value function.

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions to create accurate visual representations. For ƒ(x) = |x| - 3, one can start by plotting key points, such as the vertex at (0, -3), and then using the properties of the absolute value function to extend the graph symmetrically. Mastery of these techniques is vital for effectively visualizing and interpreting the function.