When attempting to sketch curves of functions of quotients of linear factors, it is best to start by sketching the asymptotes, which imply the behaviour of the function as one coordinate tends to infinity.
If the denominator factorises or has real roots, then there is a vertical asymptote for each root of the formEach distinct linear factor gives rise to an asymptote such that the sign of the becomesor vice versa as the asymptote is crossed.
If a linear factor is squared thedoes not change sign.
We can find– asymptotes by lettingtend to infinity and manipulating the numerator to allow terms to cancel if possible.
If the degree of the denominator is greater than the degree of the numerator the y – asymptote is For examplehas a– asymptotebecause the degree of the numerator is 1 and the degree of the denominator is 2.
If the degree of the denominator equals the degree of the numerator then the– asymptote equals the quotient of the coefficient of the highest powers ofFor examplehas a – asymptotebecause the degree of the numerator equals the degree of the denominator (both equal 2) and the coefficient ofin the numerator is 3 and the coefficient ofin the denominator is 2.
We can also find intersections with axes if these exist by settingandrespectively. Ifputtinggives the intersection with the– axis asand putting y=0 implies