Final answer:
The horizontal asymptote of the function f(x) = -2x^2+3x+6 / x^2+1 is y = -2. This is determined by considering the leading coefficients of the numerator and the denominator since both are of degree 2.
Step-by-step explanation:
To find the horizontal asymptote of the function f(x) = -2x2+3x+6 / x2+1, we need to consider the degree of the polynomial in the numerator and the degree of the polynomial in the denominator. Here, both the numerator and the denominator are polynomials of degree 2.
When the degrees are equal, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and the denominator. In this case, the leading coefficient of the numerator is -2 and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is the constant ratio of these two coefficients, which is y = -2 / 1, or y = -2.
The horizontal asymptote represents the value that the function approaches as x tends towards infinity or negative infinity. For the given function, no matter how large or small x becomes, the value of f(x) will approach the line y = -2.
Understanding the concept of horizontal asymptotes is important in graphing rational functions and analyzing their end behavior.