Graph the functions in Exercises 37–56.


y = (1/x²) − 1

Verified step by step guidance

1. Step 1: Identify the basic function and its transformation. The given function is y = (1/x²) - 1. The basic function here is y = 1/x², which is a rational function with a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
2. Step 2: Determine the transformations. The function y = (1/x²) - 1 involves a vertical shift. The '-1' indicates a downward shift of the graph by 1 unit. This means the horizontal asymptote will move from y = 0 to y = -1.
3. Step 3: Analyze the behavior of the function. As x approaches 0 from the positive or negative side, the term 1/x² becomes very large, so y approaches infinity. As x approaches infinity or negative infinity, 1/x² approaches 0, so y approaches -1.
4. Step 4: Identify key points and plot them. Calculate a few key points to help sketch the graph. For example, when x = 1, y = 0; when x = -1, y = 0; when x = 2, y = -0.75; and when x = -2, y = -0.75.
5. Step 5: Sketch the graph. Plot the asymptotes and key points on a coordinate plane. Draw the curve approaching the asymptotes, ensuring it reflects the behavior analyzed in previous steps. The graph should show two branches, one in the first quadrant and one in the second quadrant, both approaching the horizontal asymptote y = -1 as x moves away from zero.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values). Understanding how to identify key features such as intercepts, asymptotes, and the overall shape of the graph is essential for accurately representing the function.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For the function y = (1/x²) - 1, there is a horizontal asymptote at y = -1, indicating that as x approaches infinity or negative infinity, the function's value approaches -1. Recognizing asymptotic behavior is crucial for understanding the long-term behavior of the graph.

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the case of y = (1/x²) - 1, the graph of y = 1/x² is shifted downward by 1 unit. Understanding these transformations helps in predicting how changes to the function's equation affect its graph.