Section 3.7 Day 2

Summary

The lesson covers slant (oblique) asymptotes, graphing rational functions using asymptotes and intercepts, and identifying holes caused by common factors.

Slant asymptotes

A rational function has a slant asymptote when the numerator’s degree is exactly one greater than the denominator’s. Use polynomial long division or synthetic division to write the function as

[ \text{linear quotient}+\frac{\text{remainder}}{\text{denominator}}. ]

The quotient gives the slant asymptote because the remainder fraction approaches zero as (x) becomes very large in either direction.

A graph can cross a slant asymptote, just as it can cross a horizontal asymptote. It cannot cross a vertical asymptote, which marks a value where the function is undefined.

Example:

[ \frac{x^2-4x-5}{x-3}=x-1-\frac{8}{x-3}. ]

The slant asymptote is (y=x-1). The instructor also identifies the graph’s intercepts and vertical asymptote, then uses them to sketch the branches.

A calculator can help illustrate this behavior. Graph the rational function and its slant asymptote, then zoom out. Far from the vertical asymptote, the function’s graph increasingly resembles the slant line.

Graphing rational functions

The examples use this general process:

1. Factor the numerator and denominator when possible.
2. Find the x-intercepts by setting the numerator equal to zero, while checking that each value is in the domain.
3. Find the y-intercept by evaluating the function at (x=0), if defined.
4. Find vertical asymptotes from denominator zeros that do not cancel.
5. Determine the end-behavior asymptote:
   • If the numerator’s degree is greater than the denominator’s, there is no horizontal asymptote. If the degrees differ by one, find the slant asymptote by division.
   • If the denominator’s degree is greater, the horizontal asymptote is (y=0).
6. Mark the intercepts and asymptotes, then sketch the possible branches.
7. If a branch’s direction is unclear, evaluate the function at a test value in that interval. The result’s sign indicates whether the graph is above or below the (x)-axis.
8. Ensure the sketch remains a function and does not cross a vertical asymptote.

The lesson applies this process to examples with multiple vertical asymptotes, using test values on either side to determine the signs and shapes of the branches.

Holes from common factors

If the numerator and denominator share a factor, canceling it simplifies the expression. However, the canceled factor still marks a value excluded from the original function’s domain. That excluded point is a hole, not a vertical asymptote.

Example:

[ \frac{x-4}{x^2-16}=\frac{1}{x+4},\qquad x\ne4. ]

The simplified expression has a vertical asymptote at (x=-4), while the original function is undefined at (x=4). Substituting (x=4) into the simplified expression gives (y=\frac18), so the graph has a hole at (\left(4,\frac18\right)).

Speakers and sources

• Instructor/narrator: Jason Cavanaugh, as attributed to the video’s channel.
• No other speakers or external sources are featured in the subtitles.