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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 53
Capitolo 4, Problema 53

Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. h(x)=1/x2 − 4

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Identify the base function given, which is \(f(x) = \frac{1}{x^2}\). This is a rational function with a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
Look at the given function \(h(x) = \frac{1}{x^2} - 4\). Notice that it is the base function \(f(x)\) shifted vertically by subtracting 4.
Understand that subtracting 4 from \(f(x)\) shifts the entire graph downward by 4 units. This means the horizontal asymptote moves from \(y=0\) to \(y=-4\).
To graph \(h(x)\), start by sketching the graph of \(f(x) = \frac{1}{x^2}\), which has a vertical asymptote at \(x=0\) and approaches zero as \(x\) goes to infinity or negative infinity.
Then, shift every point on the graph of \(f(x)\) down by 4 units to get the graph of \(h(x)\). The vertical asymptote remains at \(x=0\), and the new horizontal asymptote is \(y=-4\).

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Parent Rational Functions

Parent rational functions like f(x) = 1/x and f(x) = 1/x² serve as the basic models for more complex rational functions. Understanding their shapes, asymptotes, and behavior helps in analyzing transformations applied to them.
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Intro to Rational Functions

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the parent function. For h(x) = 1/x² − 4, subtracting 4 shifts the graph downward by 4 units, affecting the position of the horizontal asymptote and the overall graph.
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Domain & Range of Transformed Functions

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. For rational functions like h(x) = 1/x² − 4, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the end behavior, influenced by transformations.
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Introduction to Asymptotes
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