Understanding the Graph of y x - 2

The function y x - 2 is a linear equation that describes a straight line with a slope of 1 and a y-intercept at -2. This means that as x increases by 1, y increases by 1 as well. Below, we explore the graph of this function, its symmetry, and how to approach related transformations.

The Basic Graph of y x - 2

Let's start by plotting a few points for the equation y x - 2:

When x 0, y -2. When x 1, y -1. When x 2, y 0. When x 3, y 1.

By plotting these points and connecting them, we can see that the graph of y x - 2 is a straight line that crosses the y-axis at -2 and the x-axis at 2, forming a 45-degree angle with the x-axis.

Symmetry in the Graph of y x - 2

Now, let's consider the symmetry of the graph. The line y x - 2 can be reflected about the x-axis to form another line. However, the problem states that we should draw the line and then the symmetry of the line above the x-axis. This implies a reflection of the line below the x-axis about the x-axis and keeping the line above the x-axis as is.

Procedure for Drawing the Graph

To draw the graph, follow these steps:

Draw the line y x - 2. This line has a slope of 1 and a y-intercept at -2. Reflect the portion of the line that lies below the x-axis about the x-axis. This reflection will create a mirror image of the line y x - 2 below the x-axis, which we then connect to the original line above the x-axis.

The result is a graph that consists of two linear segments forming a V-shape with its minimum point at (2, 0).

Generalizing y x - a

The equation y x - a represents a line that is parallel to the line y x - 2, but instead of having a y-intercept at -2, it has a y-intercept at -a. The graph of y x - a will always be a straight line with a slope of 1, and the distance from any point x to a is represented by |x - a|.

For example, the equation y x - 3 represents a line that is parallel to y x - 2 but with a y-intercept at -3. The graph will have a 45-degree slope and will cross the x-axis at 3.

In general, the shape described by the equation y x - a is a V-shape with its pointed point at (a, 0). This V-shape indicates that all points x are at least a distance a from the point (a, 0).

For y x - 2, this means all x values can be within the range (-∞, -1) to (5, ∞).

Drawing out the real number line can be helpful in visualizing this concept. The real number line extends infinitely in both directions, and the line y x - 2 spans from negative infinity to positive infinity, being closer to the y-axis at x 2.

Conclusion

Understanding the graph of y x - 2 and its symmetries is key to grasping linear equations and graph transformations. By following the procedure outlined, we can accurately draw and interpret these graphs, applying this knowledge to similar equations.

For more complex transformations and related problems, simply follow the same principles of slope and y-intercept. If you have further questions or need specific examples, feel free to ask!