Find the polynomial end-behavior asymptote in the graph of f(x)=(2x^3 +6x-1)/(x+2)
is it y=2x^2 -4x+14 ?
yes, that would be the asymptotic curve.
To find the end-behavior asymptote of a rational function, we need to examine the degrees of the numerator and denominator polynomials.
In the given function, the degree of the numerator is 3 (highest exponent of x), and the degree of the denominator is 1.
When the degree of the numerator is larger than the degree of the denominator by exactly 1, as in this case, the end-behavior asymptote is a slant asymptote.
To determine the equation of the slant asymptote, we perform long division of the numerator by the denominator.
Dividing (2x^3 + 6x - 1) by (x + 2):
2x^2 - 4x + 8
-----------------------
x + 2 | 2x^3 + 6x - 1
2x^3 + 4x^2
-------------
2x^2 + 6x
2x^2 + 4x
-------------
2x - 1
The result of the division is (2x^2 - 4x + 8) plus the remainder (2x - 1) divided by (x + 2).
Therefore, the equation of the slant asymptote is y = 2x^2 - 4x + 8.
Note: The slant asymptote indicates the behavior of the function as x approaches positive or negative infinity.