Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

5. Rational Functions

Graphing Rational Functions

College Algebra

5. Rational Functions

Graphing Rational Functions

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2:39 minutes

Problem 55

Textbook Question

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/(x−3)^2+1

Verified step by step guidance

Identify the base function: The base function here is \( f(x) = \frac{1}{x^2} \).

Determine the transformations: The function \( h(x) = \frac{1}{(x-3)^2} + 1 \) involves a horizontal shift and a vertical shift.

Apply the horizontal shift: The term \( (x-3) \) indicates a shift to the right by 3 units.

Apply the vertical shift: The \(+1\) outside the fraction indicates a shift upwards by 1 unit.

Graph the transformed function: Start with the graph of \( f(x) = \frac{1}{x^2} \), shift it right by 3 units, and then shift it up by 1 unit.

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Key Concepts

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

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors such as asymptotes, intercepts, and discontinuities. Understanding the structure of rational functions is essential for analyzing their graphs, particularly how they behave near their vertical and horizontal asymptotes.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function h(x) = 1/(x−3)^2 + 1 represents a vertical shift and a horizontal shift of the basic function f(x) = 1/x^2. Recognizing these transformations helps in accurately sketching the graph of the function based on its parent function.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, while horizontal asymptotes indicate the behavior of the function as x approaches infinity. Identifying these asymptotes is crucial for understanding the overall shape and limits of the graph of a rational function.