Question

Graph each function. Be sure to label three points on the graph. $$f(x)=|x|$$

Short answer

Plot points (-1,1), (0,0), and (1,1), then connect them to form a V shape.

Step-by-step solution

- Identify Key Points

Identify three key points to plot for the function. For the absolute value function, let's choose x-values: -1, 0, and 1.

- Evaluate the Function at Key Points

Calculate the value of the function at each of the chosen x-values. \ For x = -1: \ \(f(-1) = |-1| = 1\) \ For x = 0: \ \(f(0) = |0| = 0\) \ For x = 1: \ \(f(1) = |1| = 1\)

- Plot the Points

Plot the points on a coordinate plane: (-1,1), (0,0), and (1,1).

- Sketch the Graph

Connect the plotted points with straight lines to form the V-shape characteristic of the absolute value function. The lines should form a 'V' shape centered at the origin (0,0).

- Label the Points

Clearly label the three points you plotted: (-1,1), (0,0), and (1,1) on the graph. Ensure the x-axis and y-axis are also labeled.

Key concepts

Coordinate Plane

Understanding the coordinate plane is the first step to graphing any function. The coordinate plane is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
Each point on the plane can be described by an ordered pair \(x, y\), where \x\ represents the horizontal distance from the origin and \y\ represents the vertical distance. The origin is the point \(0,0\) where both axes intersect.

Here are some important aspects:

• The horizontal (x) values increase to the right and decrease to the left of the origin.
• The vertical (y) values increase when moving up and decrease when moving down from the origin.

When plotting points, it's crucial to start at the origin and move according to the ordered pair's values. This understanding lays the foundation for accurately plotting any function.

Plotting Points

Plotting points involves placing points on the coordinate plane according to their coordinates. Let's use the example from the exercise to understand this process. We have the points (-1, 1), (0, 0), and (1, 1).

• First point: (-1, 1). Start at the origin, move 1 unit left, and then 1 unit up.
• Second point: (0, 0). This is the origin, so no movement is needed.
• Third point: (1, 1). Start at the origin, move 1 unit right, and then 1 unit up.

Plotting these points is crucial for graphing the absolute value function efficiently. Once plotted correctly, connections between them will visualize the function.

V-shape Graph

The absolute value function \(f(x) = |x|\) creates a distinct V-shape graph. This shape occurs because the function outputs the distance of any x-value from zero, effectively flipping negative values to positive. Here's a closer look at the steps to sketch this graph:

• Identify and plot points: Use key points like (-1, 1), (0, 0), and (1, 1).
• Draw the lines: Connect these points with straight lines.
  This will naturally form a V-shape.
• Label your points: Ensure the x-axis and y-axis, as well as the points, are clearly labeled.

This V-shape is symmetric around the y-axis and centered at the origin. Understanding this will help in accurately sketching and interpreting absolute value functions.