Graph the functions in Exercises 37–56.


y = |x − 2|

Verified step by step guidance

1. Identify the basic form of the function: The given function is \( y = |x - 2| \), which is an absolute value function. The graph of an absolute value function \( y = |x - a| \) is a V-shaped graph with its vertex at the point \( (a, 0) \).
2. Determine the vertex: For the function \( y = |x - 2| \), the vertex is at \( (2, 0) \). This is because the expression inside the absolute value, \( x - 2 \), equals zero when \( x = 2 \).
3. Analyze the slope of the lines: The graph of \( y = |x - 2| \) consists of two linear pieces. For \( x < 2 \), the function behaves like \( y = -(x - 2) = -x + 2 \), which has a negative slope. For \( x > 2 \), the function behaves like \( y = x - 2 \), which has a positive slope.
4. Plot key points: Start by plotting the vertex at \( (2, 0) \). Then, choose points on either side of the vertex to plot. For example, for \( x = 1 \), \( y = |1 - 2| = 1 \), and for \( x = 3 \), \( y = |3 - 2| = 1 \). Plot these points: \( (1, 1) \) and \( (3, 1) \).
5. Draw the graph: Connect the points with straight lines to form the V-shape. The left side of the V will have a negative slope, and the right side will have a positive slope, both meeting at the vertex \( (2, 0) \). This completes the graph of the function \( y = |x - 2| \).

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. For the function y = |x - 2|, it represents the distance between x and 2, creating a V-shaped graph that opens upwards with its vertex at the point (2, 0).

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize mathematical functions. For y = |x - 2|, one can identify key points, such as the vertex and intercepts, and use symmetry to sketch the graph accurately. Understanding how to plot these points is essential for creating a clear representation of the function.

Vertex of a Function

The vertex of a function is the point where the graph changes direction, often representing a minimum or maximum value. In the case of y = |x - 2|, the vertex is located at (2, 0), which is the lowest point of the V-shaped graph. Recognizing the vertex helps in understanding the overall shape and behavior of the function.