Overview
This lecture covers how to find horizontal, vertical, and slant (oblique) asymptotes for rational functions, with step-by-step examples including analysis of graph behavior, domain, and range.
Vertical Asymptotes
- Set the denominator equal to zero and solve for ( x ) to find vertical asymptotes.
- The value(s) where the denominator is zero (and not canceled by the numerator) is the vertical asymptote.
Horizontal Asymptotes
- Compare degrees of numerator and denominator:
- If denominator degree > numerator (bottom heavy), horizontal asymptote is ( y = 0 ).
- If degrees are equal, horizontal asymptote is ratio of leading coefficients.
- If numerator degree > denominator (top heavy), there is no horizontal asymptote.
Slant (Oblique) Asymptotes
- Occur when numerator degree is exactly one more than denominator.
- Use long division; slant asymptote is the linear quotient (ignore remainder).
Holes (Point Discontinuities)
- If a factor cancels in numerator and denominator, the corresponding ( x )-value is a hole.
- To find hole’s ( y )-coordinate, substitute ( x ) into the simplified function.
Domain and Range
- Domain excludes vertical asymptote(s) and ( x )-values of holes.
- Range excludes horizontal asymptote(s) and hole’s ( y )-value(s).
Example Summaries
Example 1: ( y = \frac{1}{x-3} )
- Vertical asymptote: ( x = 3 ).
- Horizontal asymptote: ( y = 0 ).
- Domain: ( (-\infty, 3) \cup (3, \infty) ).
- Range: ( (-\infty, 0) \cup (0, \infty) ).
Example 2: ( y = \frac{1}{x+2} + 7 )
- Vertical asymptote: ( x = -2 ).
- Horizontal asymptote: ( y = 7 ).
- Domain: ( (-\infty, -2) \cup (-2, \infty) ).
- Range: ( (-\infty, 7) \cup (7, \infty) ).
Example 3: ( y = \frac{6x-18}{2x+4} )
- Vertical asymptote: ( x = -2 ).
- Horizontal asymptote: ( y = 3 ).
- X-intercept: ( x = 3 ).
- Y-intercept: ( y = -4.5 ).
- Domain: ( (-\infty, -2) \cup (-2, \infty) ).
- Range: ( (-\infty, 3) \cup (3, \infty) ).
Example 4: ( y = \frac{2x^2 - 3x - 2}{x^2 + x - 6} )
- Factor to find vertical asymptote(s) and hole(s).
- Vertical asymptote: ( x = -3 ).
- Hole at ( x = 2 ), ( y = 1 ).
- Horizontal asymptote: ( y = 2 ).
- Domain: ( (-\infty, -3) \cup (-3, 2) \cup (2, \infty) ).
- Range: ( (-\infty, 1) \cup (1, 2) \cup (2, \infty) ).
Example 5: ( y = \frac{2x^2-x+1}{x-2} )
- Vertical asymptote: ( x = 2 ).
- No horizontal asymptote (numerator degree > denominator).
- Slant asymptote: ( y = 2x + 3 ).
Key Terms & Definitions
- Rational Function — A function of the form ( \frac{P(x)}{Q(x)} ) where ( P ) and ( Q ) are polynomials.
- Vertical Asymptote — Vertical line ( x = a ) where function approaches infinity.
- Horizontal Asymptote — Horizontal line ( y = b ) showing end behavior as ( x \to \pm\infty ).
- Slant/Oblique Asymptote — A non-horizontal asymptote occurring when the numerator's degree is one higher than the denominator’s.
- Hole (Point Discontinuity) — A point where both numerator and denominator are zero at the same ( x )-value after factoring.
Action Items / Next Steps
- Practice factoring rational functions and identifying all asymptotes and holes.
- Graph several rational functions, marking asymptotes, holes, and intercepts.
- Review polynomial long division for finding slant asymptotes.