3.4 Sketching and interpreting rational functions

A rational graph is assembled from domain restrictions, discontinuities, intercepts, sign intervals, and end behaviour. Each feature constrains the final sketch.

3.4.1 A systematic graphing process

First factor and state the domain. Record holes before cancelling. Find intercepts, vertical asymptotes, and one-sided behaviour. Use zeros and asymptotes to divide the number line into sign intervals. Determine horizontal behaviour from degrees, then sketch branches that respect every feature.

Worked Example: Graphing a rational function with two asymptotes

For \(f(x)=(2x-1)/(x+2)\), the domain excludes \(-2\), the \(x\)-intercept is \(1/2\), the \(y\)-intercept is \(-1/2\), the vertical asymptote is \(x=-2\), and the horizontal asymptote is \(y=2\). Test points determine the sign of each branch.

Try It. List the key graphing features of \((x+3)/(x-1)\).
Check Your Work Domain \(x\ne1\); \(x\)-intercept \(-3\); \(y\)-intercept \(-3\); vertical asymptote \(x=1\); horizontal asymptote \(y=1\).

Worked Example: Graphing with a hole

For \(r(x)=(x^2-4)/(2x^2+2x)\),

\[r(x)=\frac{(x-2)(x+2)}{2x(x+1)}.\]

No factor cancels, so there is no hole. The vertical asymptotes are \(x=0,-1\), zeros are \(x=\pm2\), and the horizontal asymptote is \(y=1/2\).

Try It. Does \((x^2-1)/(x^2+3x+2)\) contain a hole?
Check Your Work Yes. It simplifies from \((x-1)(x+1)/[(x+1)(x+2)]\) and has a hole at \(x=-1\).

3.4.2 Domain and range from a rational graph

Domain is read horizontally and excludes holes and vertical asymptotes. Range is read vertically and may exclude a horizontal value or the output at a hole. Solving \(y=f(x)\) for \(x\) can help identify excluded output values when the graph is difficult to read.

Worked Example: Finding the range of a reciprocal transformation

For \(f(x)=3+2/(x-4)\), the vertical asymptote is \(x=4\) and the horizontal asymptote is \(y=3\). Since \(2/(x-4)\) can never be zero, the range is \((-\infty,3)\cup(3,\infty)\).

Try It. Find the domain and range of \(-1+5/(x+2)\).
Check Your Work Domain excludes \(-2\); range excludes \(-1\).

Worked Example: Checking a physically restricted branch

If \(H(q)=40q/(q+5)\) models head for \(q\ge0\), the algebraic vertical asymptote \(q=-5\) lies outside the physical domain. On \(q\ge0\), \(H\) begins at 0, increases, and approaches 40 from below.

Try It. Interpret \(C(t)=10t/(t+2)\) for \(t\ge0\).
Check Your Work It begins at 0, increases for nonnegative time, and approaches 10 without reaching it.

Applied Problem: Analyzing a saturating treatment model. Removal efficiency is modelled by \(E(C)=90C/(C+4)\) for dosage \(C\ge0\). Find the intercept, identify the relevant asymptotes, describe the physical range, and calculate \(E(8)\).

Check Your Work The intercept is \((0,0)\). The algebraic vertical asymptote \(C=-4\) is outside the physical domain. The horizontal asymptote is \(E=90\), so the physical range is \([0,90)\). At \(C=8\), \(E=90(8)/12=60\).

3.4.3 Practice Problems

  1. List the intercepts and asymptotes of \((x-2)/(x+1)\).
    Check Your Work \(x\)-intercept 2; \(y\)-intercept \(-2\); vertical asymptote \(x=-1\); horizontal asymptote \(y=1\).
  2. Determine the sign of \((x-3)/(x+2)\) on \((-2,3)\).
    Check Your Work Negative.
  3. Find the domain and range of \(2+1/(x-5)\).
    Check Your Work Domain excludes 5; range excludes 2.
  4. Identify all graphing features of \(x/(x^2-4)\).
    Check Your Work Domain excludes \(\pm2\); intercept \((0,0)\); vertical asymptotes \(x=\pm2\); horizontal asymptote \(y=0\).
  5. Explain why a graph cannot pass through a hole.
    Check Your Work The corresponding input is excluded from the original domain.
  6. A dosage response is \(E(C)=75C/(C+3)\) for \(C\ge0\). Find \(E(6)\) and the limiting efficiency.
    Check Your Work \(E(6)=50\); the limiting efficiency is 75.
  7. A hydraulic index is \(H(q)=20q/(q+10)\) for \(q\ge0\). State its physical range.
    Check Your Work \([0,20)\).
  8. A model \(P(q)=(q-5)/(q+1)\) is used only for \(q\ge0\). Which algebraic features lie outside the physical domain?
    Check Your Work The vertical asymptote \(q=-1\) lies outside the physical domain.