Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Section 3.6 - Rational Expressions - 3.6 Exercises - Page 309: 46

Answer

X-intercept: None Y-intercept: (0, -9/4) Vertical asymptote: x = -2 Domain: (-∞, -2) U (-2, ∞) Horizontal asymptote: y=-2 Range: (-∞, -2) Graph is below

Work Step by Step

$r(x)=\frac{-2x^2 - 8x - 9}{x^2 + 4x + 4}$ -2x^2 - 8x - 9= 0 $2 (x^2 + 4x + 4) = -9 + 8$ $2 (x+2)^2 = -1$ No x-intercept y-intercept is the ratio of the constants, which is -9/4 Thus, the y-intercept is at (0, -9/4) Vertical asymptotes are when the denominator is equal to 0 $x^2 + 4x + 4 = 0$ $(x+2)^2 = 0$ x = -2 So, domain is from (-∞, -2) U (-2, ∞) Horizontal asymptote is the ratio of the constants of the leading term (with equal degree) -2/1 = -2 Thus, the horizontal asymptote is at y=-2 So the range is from (-∞, -2)
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