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

step 1 Okay, in this problem, we want to graph the function H of x equals 1 over x minus 3 squared plus

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

Key Concepts

Reciprocal Function

A reciprocal function, typically expressed as f(x) = 1/x, is characterized by a hyperbolic graph with two distinct branches. These functions have vertical and horizontal asymptotes which serve as boundaries that the graph will approach but never cross, defining key features such as domain restrictions and behavior near zero.

Function Transformations

Function transformations involve modifying a base function by shifting, stretching, compressing, or reflecting it. In the context of reciprocal functions, these transformations allow one to reposition the asymptotes and alter the graph’s shape while preserving the essential reciprocal behavior.

Horizontal Shifts

Horizontal shifts occur when the input variable is modified by a constant, for example, replacing x with (x - 1). This results in the graph moving to the right or left. For reciprocal functions, a horizontal shift moves the vertical asymptote accordingly, changing the point around which the function's values are excluded.

Vertical Shifts

Vertical shifts involve adding or subtracting a constant to the entire function, which translates the graph up or down. For a reciprocal function, such a shift moves the horizontal asymptote, thereby changing the baseline level that the function approaches as the input grows large in magnitude.

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