LEMMA

Advanced functions

Rational functions and asymptotes

ALEKS placement

Zeros of the denominator become vertical asymptotes; the degree comparison sets the horizontal one.

What this covers

  • Factor top and bottom and cancel any shared factor — that becomes a hole, not an asymptote.
  • Set the remaining denominator factors to zero for vertical asymptotes.
  • Bottom degree larger gives y = 0; equal degrees give the ratio of leading coefficients; top larger by one gives a slant asymptote.
  • Zeros of the numerator are the x-intercepts.

Worked example

Worked example

Find the asymptotes of f(x) = (2x² + 1)/(x² - 4).

  1. x=2,x=−2x² - 4 = (x - 2)(x + 2) is zero there and nothing cancels.
  2. y=2Equal degrees, so take the ratio 2/1.
Another worked example

Analyze f(x) = [(x − 1)(x + 2)]/[(x − 1)(x − 3)].

  1. f(x)=x−3x+2​,x=1,3Cancel one common factor while keeping both original exclusions.
  2. hole (1,−23​),vertical asymptote x=3The reduced formula has a finite value at 1 but still has a denominator zero at 3.
  3. horizontal asymptote y=1,x-intercept (−2,0)Equal degrees give the ratio of leading coefficients. The numerator zero is allowed here.

A common mistake

A mistake Lemma catches

From x=2,x=−2

y=−41​y=2

“Far from 0 the highest powers take over: (2x² + 1)/(x² − 4) behaves like 2x²/x², so the graph levels off at 2/1 = 2. The constants 1 and −4 matter less and less as x grows; their ratio −1/4 is the height at x = 0, not where the graph levels off.”

Try one

Sample problem

Enter the x-value of the vertical asymptote.

f(x)=x−1x+3​

Practice rational functions free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this