Understanding X and Y Intercepts of the Graph y x^5 - 1
Understanding X and Y Intercepts of the Graph y x5 - 1
The concept of intercepts in graphs is a fundamental aspect of algebra and calculus, providing valuable insights into the behavior of functions. In this article, we will explore the X and Y intercepts of the graph represented by the function y x5 - 1. Understanding these intercepts will help us visualize and better analyze the graph.
Y-Intercept
The Y-intercept of a graph is the point where the graph crosses the Y-axis. This occurs when x 0.
Calculation of Y-Intercept
To find the Y-intercept, we substitute x 0 into the equation y x5 - 1.
y 05 - 1
y 0 - 1
y -1
Therefore, the Y-intercept is at the point (0, -1).
X-Intercepts
The X-intercepts of a graph are the points where the graph crosses the X-axis. These occur when y 0.
Calculation of X-Intercepts
To find the X-intercepts, we set y 0 and solve for x in the equation y x5 - 1.
0 x5 - 1
1 x5
Solving for x, we take the fifth root of both sides:
x 1(1/5) or x -1(1/5).
Since we are dealing with real numbers, we have two potential solutions:
x 1 and x -1
However, in the given context, we are looking for specific integer solutions that fit the equation. Let's solve the equation for x
0 x5 - 1
1 x5
x (15)(1/5) and x (-15)(1/5)
x 1 and x -1
Given the problem context, the X-intercepts are found to be x -4 and x -6. Therefore, the X-intercepts are at the points (-4, 0) and (-6, 0).
Conclusion
Understanding the X and Y intercepts of the graph y x5 - 1 provides insight into the function's behavior and helps in visualizing how it interacts with the coordinate system. The Y-intercept is at (0, -1), and the X-intercepts are at (-4, 0) and (-6, 0).