In Exercises 69-70, use transformations of f(x)=(1)/(x) or f(x)=(1)/(x^2) to graph each rational

step 1 The goal here is to graph the function G of x equals 1 over x plus 2 squared using transformatio

In Exercises 69-70, use transformations of f(x)=(1)/(x) or...

Key Concepts

Asymptotes in Graphing Rational Functions

Asymptotes are lines that the graph of a function approaches but never touches. Transformations of functions, like horizontal and vertical shifts, also shift the positions of these asymptotes, which is important for accurately depicting the behavior of rational functions.

Parent Functions

Parent functions such as f(x) = 1/x and f(x) = 1/x² serve as the basic templates for graphing more complex rational functions. Understanding their characteristics, including their asymptotic behavior, is crucial for applying transformations correctly.

Function Transformations

This concept involves altering the base or parent function using operations like shifting, reflecting, stretching, or compressing. It serves as the foundation for modifying known graphs to fit new functions, allowing one to understand how changes in the algebraic expression directly affect the graph’s appearance.

Horizontal Shifts

A horizontal shift occurs when a constant is added to or subtracted from the variable inside the function. This moves the graph left or right without changing its shape, which is essential when the function includes an expression like (x + 2) that shifts the graph to the left by 2 units.

Vertical Shifts

Vertical shifts are observed when a constant is added to or subtracted from the entire function. This moves the graph up or down. In the given function, subtracting 1 at the end of the expression shifts the graph downward by 1 unit.

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