Further rational functions

Factor numerator and denominator first, retaining every original exclusion. A remaining denominator zero usually produces a vertical asymptote; a cancelled factor can produce a hole.

Compare degrees for horizontal asymptotes; divide a quadratic numerator by a linear denominator for an oblique asymptote. Sketch each branch separately and check its approach on both sides of every vertical asymptote.

Planning a rational sketch

For a linear numerator over a quadratic denominator, factor the denominator over the reals. Two distinct real roots can give two vertical asymptotes; a repeated root can give the same infinite direction on both sides; no real roots means no vertical asymptote. After checking cancellation, the horizontal asymptote is \(y=0\).

For a quadratic numerator over a linear denominator, polynomial division gives \(\mathrm f(x)=mx+c+\dfrac{r}{dx+e}\). When \(r\ne0\), the oblique asymptote is \(y=mx+c\). When \(r=0\), the graph is that line with the original excluded input retained as a missing point.

On each interval between poles and zeros, use a test input to check the sign of the function. Evaluate one-sided behaviour at every pole. A horizontal asymptote describes end behaviour and may be crossed at a finite input; it is not a barrier.

Use graphing technology to locate turning points when required. Adjust the window to show every branch and compare with the algebraic intercepts and asymptotes. A steep segment drawn across a pole is a plotting artefact, not part of the curve.

To find a possible range algebraically, set \(y=k\) and rearrange into an equation in \(x\). Require a real permitted solution. A discriminant condition is useful when the resulting equation is quadratic, but check separately any value of \(k\) that makes its quadratic coefficient zero and reject original denominator exclusions.