Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. As _____
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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.
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.
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.