How to Graph Rational Expressions? (+FREE Worksheet!)

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Graphing Rational Expressions – Example 2:

Graph rational expressions. \(f(x)=\frac{3x}{x^2-2x}\)

Tutor-style math help

Graph Rational Expressions: what to notice and how to work it

Rational skill

Graphing a rational function means finding the places the graph cannot go and the lines it approaches. Restrictions, asymptotes, intercepts, and holes shape the sketch before you plot extra points.

What to notice first

Factor first. Denominator zeros usually lead to vertical asymptotes or holes, and the degrees of numerator and denominator guide the horizontal or slant asymptote.

Common student mistake

Do not cancel a factor and then forget it. A canceled denominator factor creates a hole, while an uncanceled denominator zero creates a vertical asymptote.

Key formulas and cues

\(f(x)=\frac{p(x)}{q(x)},\ q(x)\ne0\)

\(\text{vertical asymptote: uncanceled denominator}=0\)

\(\text{hole: canceled denominator factor}=0\)

\(\text{horizontal asymptote depends on degrees}\)

A reliable path

State restrictionsFind values that make original denominators zero.
Factor and simplifyCancel only factors shared by the whole numerator and denominator.
Check the resultKeep original restrictions and watch for asymptotes or holes when graphing.

Worked examples

Find asymptotes first

Example: \(f(x)=\frac{x+1}{x-2}\)

The denominator is zero at x = 2.
No factor cancels, so x = 2 is a vertical asymptote.
The degrees match, so the horizontal asymptote is the ratio of leading coefficients.

Answer: Vertical asymptote \(x=2\); horizontal asymptote \(y=1\).

Spot a hole

Example: \(g(x)=\frac{x^2-9}{x-3}\)

Factor the numerator: (x – 3)(x + 3).
The factor x – 3 cancels.
The graph has a hole where x = 3, not a vertical asymptote.

Answer: Hole at \(x=3\); simplified rule \(y=x+3\), so the missing point is \((3,6)\).

Open pop-up practice

Try one before moving on

Try: Find the vertical asymptote of \(h(x)=\frac{x-1}{x+6}\).

Answer: \(x=-6\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Solution:

First, notice that the graph is in two pieces. Find \(y\)-intercept by substituting zero for \(x\) and solving for \(y (f(x)): x=0→y=\frac{3x}{x^2-2x}=\frac{3(0)}{(0^2-2(0)}=\frac{0}{0}\), \(y\)-intercept: None Asymptotes of \(\frac{3x}{x^2-2x}\): vertical: \(x=2\), Horizontal: \(y=0\) After finding the asymptotes, you can plug in some values for \(x\) and solve for \(y\). Here is the sketch for this function.