Rational functions, asymptotes and curve sketching

Learning objectives

Locate the vertical asymptotes of a rational function at the zeros of its denominator, and find the horizontal or oblique asymptote from the behaviour as x → ±∞, obtaining an oblique asymptote by algebraic division

Sketch the graph of a rational function whose numerator and denominator are linear or quadratic, showing every asymptote, the axis intercepts, any stationary points and the direction in which the curve approaches each asymptote from either side

Determine the set of values a rational function can take, either by rearranging y = f(x) as a quadratic in x and requiring the discriminant to be non-negative, or by finding the stationary values

Sketch y² = f(x) from the graph of y = f(x), noting that the new curve exists only where f(x) ≥ 0, is symmetric in the x-axis, meets it where f(x) = 0 and has a vertical tangent there

Sketch y = 1/f(x), y = |f(x)| and y = f(|x|) from the graph of y = f(x), explaining how maxima of f become minima of 1/f, how zeros of f become vertical asymptotes of 1/f, how negative parts are reflected in the x-axis for |f(x)|, and how the part of the curve for x ≥ 0 is reflected in the y-axis for f(|x|)

Use a completed sketch to state the number of real roots of an equation or to solve an inequality involving a rational expression, giving the answer as a union of intervals