Understanding the Graph of y x6: A Comparative Analysis with y x

The question at hand involves the drawing and analysis of graphs for the functions y x6 and y x.

Graphing y x and y x6

To begin, let's consider the simpler linear function y x, which we will call Graph B.

Graph B: y x

When x 0, it is clear that y 0. This means that Graph B passes through the origin (0,0) and is a straight line extending in both directions with a slope of 1.

Graphing y x6

Next, we focus on the function y x6, which we will denote as Graph A.

When x 0, y 06 0. However, the key difference arises when x ≠ 0. Since x6 is an even power, all values of x (both positive and negative) will yield a positive result for y. This means the graph is symmetric about the y-axis and will only involve positive y-values for both positive and negative x-values.

To further illustrate, when x 1, y 16 1, and when x -1, y (-1)6 1. Therefore, the graph of y x6 will pass through the origin and extend upwards in the first and second quadrants, and downwards in the third and fourth quadrants.

Similitude Between Graph A and Graph B

There is an interesting observation to be made when comparing Graph A and Graph B. Looking at the values of y for both cases when x 0, we see that Graph A and Graph B both yield y 0. However, the behavior of these functions for other values of x is quite different.

For x 0, in Graph B, y x, resulting in a downward-sloping line, while in Graph A, y x6, which results in a flat line at y 0 and increases rapidly in the negative x-direction. Conversely, for x 0, both Graph A and Graph B increase linearly, but Graph A does so at a rate determined by the sixth power.

Shifting Graph B to Form Graph A

Given this, it is evident that Graph A is similar to Graph B, but with a key difference in y-values for positive x-values. Specifically, the minimum value of y in Graph A is 6, whereas in Graph B, it is 0. This indicates that the entire Graph B can be thought of as a shifted version of Graph A, but in the opposite direction.

To put it more concretely, the graph of y x6 (Graph A) appears to be the same as y x (Graph B), but with an upward shift of 6 units. This is because the y-values for Graph B at each point are 6 units lower than those of Graph A.

Conclusion

In conclusion, the graph of y x6 involves a transformation of the graph of y x, resulting in a compilation of positive y-values with a significant upward shift of 6 units. This demonstrates how even and odd power functions behave differently and can be related through simple transformations.

Keywords

graph of y x?, y x, comparative analysis, graph shifting, function transformation