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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 19
Chapter 4, Problem 19

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
Graph of a rational function with vertical asymptotes at x = -2 and x = 1, and horizontal asymptote at y = 1.
As x→∞, f(x)→x\(\to\]\infty\),\(\text{ }\)f(x)\(\to\)_____

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1
Identify the horizontal asymptote from the graph. The horizontal asymptote is the line that the function approaches as x approaches infinity or negative infinity.
From the graph, observe the horizontal dashed line labeled as the horizontal asymptote, which is at y = 17.
Recall that for rational functions, the horizontal asymptote represents the value that f(x) approaches as x approaches infinity (x → ∞) or negative infinity (x → -∞).
Therefore, as x → ∞, the function f(x) approaches the horizontal asymptote y = 17.
Write the conclusion: As x → ∞, f(x) → 17.

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator is zero and the function approaches infinity or negative infinity. They represent values of x where the function is undefined and the graph shows a vertical line that the curve approaches but never crosses.
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Determining Vertical Asymptotes

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote is a horizontal line y = c that the graph approaches, indicating the end behavior of the function.
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Determining Horizontal Asymptotes

End Behavior of Rational Functions

The end behavior of a rational function is determined by the degrees of the numerator and denominator polynomials. It shows how the function behaves as x approaches positive or negative infinity, often approaching a horizontal asymptote or increasing/decreasing without bound.
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End Behavior of Polynomial Functions
Related Practice
Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=x3−x2−9x+9f(x) = x^3 - x^2 - 9x + 9

Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=11x3−6x2+x+3f(x)=11x^3−6x^2+x+3

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Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x3+7x2−x+9f(x)=5x^3+7x^2−x+9

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Textbook Question

Divide using synthetic division. (3x2+7x−20)÷(x+5)

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Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=4x−x3f(x) = 4x - x^3

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Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x3−10x−12=0