Algebra 2 Honors Practice Test

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Find the oblique asymptote of R(x) = (2x^2 + x + 1)/(x + 1) and describe the end behavior.

y = 2x − 1; as x → ±∞, R(x) ~ 2x − 1

When the top degree is one higher than the bottom, an oblique (slant) asymptote is given by the quotient from dividing the numerator by the denominator. Do the division: (2x^2 + x + 1) ÷ (x + 1). The quotient is 2x − 1 with remainder 2, so

R(x) = (2x^2 + x + 1)/(x + 1) = 2x − 1 + 2/(x + 1).

As x grows large in either direction, the term 2/(x + 1) tends to 0, so R(x) behaves like the line y = 2x − 1. Therefore the oblique asymptote is y = 2x − 1, and the end behavior is R(x) ~ 2x − 1 as x → ±∞. (Note there’s a vertical boundary at x = −1 from the 2/(x+1) term, but it doesn’t affect the oblique asymptote.)

y = x; as x → ∞, R(x) ~ x

y = 2x + 1; as x → ∞, R(x) ~ 2x + 1

No oblique asymptote; horizontal asymptote y=0