Overview
The transcript explains how to find and graph vertical, horizontal, and slant (oblique) asymptotes of rational functions, along with domain, range, intercepts, holes, and end behavior.
Asymptotes: Concepts and Rules
- Vertical asymptote (VA): Set denominator equal to zero; exclude any factor that cancels with numerator.
- Horizontal asymptote (HA):
- Bottom heavy (deg numerator < deg denominator): y = 0.
- Same degree: y = ratio of leading coefficients.
- Top heavy (deg numerator > deg denominator): no HA; if degree difference is 1, a slant asymptote exists.
- Slant/oblique asymptote (SA): Use polynomial long division when deg(numerator) = deg(denominator) + 1; quotient (without remainder) gives y = mx + b.
- End behavior: y approaches the horizontal asymptote (or follows the slant asymptote) as x → ±∞.
Worked Example 1: y = 1/(x − 3)
- Vertical asymptote: x = 3.
- Horizontal asymptote: y = 0 (bottom heavy).
- End behavior: y → 0 as x → ±∞.
- Graph placement via test points:
- x = 4 → y = 1 → right branch above x-axis.
- x = 2 → y = −1 → left branch below x-axis.
- Domain: (−∞, 3) ∪ (3, ∞).
- Range: (−∞, 0) ∪ (0, ∞).
Worked Example 2: y = 1/(x + 2) + 7
- Vertical asymptote: x = −2.
- Horizontal asymptote: y = 7 (y = 0 shifted up by 7).
- Graph placement via test points:
- x = −1 → y = 8 → right of VA, above HA.
- x = −3 → y = −6 → left of VA, below HA.
- Domain: (−∞, −2) ∪ (−2, ∞).
- Range: (−∞, 7) ∪ (7, ∞).
Worked Example 3: y = (6x − 18)/(2x + 4)
- Simplification/factoring: y = [6(x − 3)]/[2(x + 2)].
- Horizontal asymptote: y = 3 (same degree; 6x/2x = 3).
- Vertical asymptote: x = −2.
- Intercepts:
- x-intercept: set numerator 0 → x = 3 → (3, 0).
- y-intercept: x = 0 → y = −18/4 = −4.5 → (0, −4.5).
- Placement check: x = −3 → y = 18 (above HA on left).
- Domain: (−∞, −2) ∪ (−2, ∞).
- Range: (−∞, 3) ∪ (3, ∞).
Worked Example 4: y = (2x^2 − 3x − 2)/(x^2 + x − 6)
- Factor:
- Denominator: x^2 + x − 6 = (x + 3)(x − 2).
- Numerator: 2x^2 − 3x − 2 = (2x + 1)(x − 2) via grouping.
- Cancellation and hole:
- Cancel (x − 2); hole at x = 2.
- Hole coordinate: plug x = 2 into surviving y = (2x + 1)/(x + 3) → y = 1 → hole at (2, 1).
- Horizontal asymptote: y = 2 (same degree; 2x^2/x^2).
- Vertical asymptote: x = −3 (surviving denominator factor).
- Domain: (−∞, −3) ∪ (−3, 2) ∪ (2, ∞) (remove VA and hole x-value).
- Range: (−∞, 1) ∪ (1, 2) ∪ (2, ∞) (remove hole y-value and HA).
Worked Example 5: y = (2x^2 − x + 1)/(x − 2)
- Vertical asymptote: x = 2 (denominator zero; numerator not factorable to cancel).
- Horizontal asymptote: none (top heavy).
- Slant asymptote via long division:
- Quotient: 2x + 3 with remainder 7 → SA: y = 2x + 3.
- Placement via points:
- x = 0 → y = 1/(−2) = −0.5 → left branch below SA.
- x = 3 → y = 16 → right branch above SA.
Summary Table of Examples
| Function | VA | HA | SA | Hole(s) | Key Points | Domain | Range |
|---|
| y = 1/(x − 3) | x = 3 | y = 0 | none | none | (4, 1), (2, −1) | (−∞, 3) ∪ (3, ∞) | (−∞, 0) ∪ (0, ∞) |
| y = 1/(x + 2) + 7 | x = −2 | y = 7 | none | none | (−1, 8), (−3, −6) | (−∞, −2) ∪ (−2, ∞) | (−∞, 7) ∪ (7, ∞) |
| y = (6x − 18)/(2x + 4) | x = −2 | y = 3 | none | none | (3, 0), (0, −4.5), (−3, 18) | (−∞, −2) ∪ (−2, ∞) | (−∞, 3) ∪ (3, ∞) |
| y = (2x^2 − 3x − 2)/(x^2 + x − 6) | x = −3 | y = 2 | none | (2, 1) | Surviving y = (2x + 1)/(x + 3) | (−∞, −3) ∪ (−3, 2) ∪ (2, ∞) | (−∞, 1) ∪ (1, 2) ∪ (2, ∞) |
| y = (2x^2 − x + 1)/(x − 2) | x = 2 | none | y = 2x + 3 | none | (0, −0.5), (3, 16) | (−∞, 2) ∪ (2, ∞) | not specified |
Key Terms & Definitions
- Vertical asymptote (VA): A vertical line x = a where the function grows without bound near x = a.
- Horizontal asymptote (HA): A horizontal line y = b that the function approaches as x → ±∞.
- Slant (oblique) asymptote (SA): A non-horizontal line y = mx + b approached by the function as x → ±∞.
- Bottom heavy: deg(numerator) < deg(denominator); HA is y = 0.
- Top heavy: deg(numerator) > deg(denominator); no HA; if degree difference is 1, SA exists.
- Hole (removable discontinuity): A point (a, b) removed from the graph due to a common factor cancellation.
- End behavior: The limiting value or line the function approaches as x → ±∞.
Action Items / Next Steps
- Practice identifying degree relationships to choose HA or SA quickly.
- Factor numerators and denominators fully to detect holes before assigning VAs.
- Use long division for top-heavy cases with degree difference of one to find SA.
- Confirm graph branch locations with test points on each side of the VA.