1.2 Rational functions and graphs

For a rational function, factorise numerator and denominator first. A denominator zero may create a vertical asymptote or a removable hole; the degrees and leading coefficients determine horizontal or oblique behaviour.

Find excluded x-values, intercepts and the sign of each factor on intervals. Long division gives the polynomial asymptote when the numerator degree is at least the denominator degree.

For (x+1)/(x−2), x=2 is a vertical asymptote, x=−1 is the x-intercept, and y=1 is the horizontal asymptote because the leading coefficients match.

A cancelled factor creates a hole rather than a vertical asymptote, and an asymptote is a limiting feature—not necessarily a line the graph can never cross in every case.

Graph transformations change coordinates systematically: y=f(x)+a shifts vertically, y=f(x−a) shifts right, y=−f(x) reflects in the x-axis, and y=f(−x) reflects in the y-axis.

For y=af(bx+c)+d, apply the inside transformation before the outside one: horizontal scale/reflection and shift act on x, while a and d act on y. Track a known point to check the direction.

A point (2,3) on y=f(x) becomes (5,3) on y=f(x−3), because the input required to produce the same output is three larger.

The sign inside f changes the horizontal direction opposite to the visible sign outside; y=f(x−3) shifts right, not left.