Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. h(x)=(1/x) + 2

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 47

Chapter 4, Problem 47

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=(1/x) + 2

Verified step by step guidance

Identify the base function given, which is \( f(x) = \frac{1}{x} \). This is a rational function with a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Look at the given function \( h(x) = \frac{1}{x} + 2 \). Notice that it is the base function \( \frac{1}{x} \) plus 2, which means the graph of \( f(x) \) is shifted vertically.

Understand that adding 2 to the function shifts the entire graph up by 2 units. This means the horizontal asymptote, originally at \( y = 0 \), will move to \( y = 2 \).

The vertical asymptote remains unchanged at \( x = 0 \) because the transformation does not affect the denominator.

To sketch the graph, start with the graph of \( f(x) = \frac{1}{x} \), then shift every point up by 2 units, and draw the new horizontal asymptote at \( y = 2 \) while keeping the vertical asymptote at \( x = 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parent Rational Functions

The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, is essential as they serve as the starting point for transformations.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For h(x) = 1/x + 2, the '+ 2' indicates a vertical shift upward by 2 units, moving the entire graph and its horizontal asymptote accordingly.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. For rational functions like 1/x, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe end behavior. Transformations affect the position of these asymptotes.

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