1. Rational Functions

2. Rational Functions a: b: h:
𝑓 𝑥 = 𝑔(𝑥) ℎ(𝑥) or 𝑓 𝑥 = 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙
Parent Function:
𝑓 𝑥 = 1 𝑥
Transformations: a:
Vertical Stretch/Shrink, and vertical reflection b:
Horizontal Stretch/Shrink, and horizontal reflection h:
Horizontal Shift k:
Vertical Shift

3. 𝑓 𝑥 = 𝑔(𝑥) ℎ(𝑥) or 𝑓 𝑥 = 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙
Rational Functions
𝑓 𝑥 = 𝑔(𝑥) ℎ(𝑥) or 𝑓 𝑥 = 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙
General Form:
𝑓 𝑥 = 𝑎 𝑏(𝑥−ℎ) +𝑘
Examples:

4. Rational Functions Domain There are two things that restrict domain
- negative values as the radicand and
- a denominator value of 0 (before it becomes recognized as a point discontinuity)
Middle Behavior The limit notation description of the behavior of the function as it approaches the vertical asymptotes and/or point discontinuities
End Behavior The limit notation description of the behavior of the function as it approaches ∞ 𝑜𝑟 −∞

5. Rational Functions
x-intercepts (zeroes) These occur at the real zeroes of the numerator (assuming they are NOT zeroes of the denominator)
y-intercept Let x = 0 and you’ll find your y.
Vertical Asymptotes These occur at the real zeroes of the denominator (assuming they are NOT zeroes of the numerator with equal or greater multiplicity, if this is the case we call them Point Discontinuities)

6. Rational Functions Horizontal Asymptote There are 3 possibilities.
- If the degree of the numerator is less than the degree of the denominator, then your H.A. is at y = 0.
- If the degree of the numerator is equal to the degree of the denominator, then your H.A. is at
y = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 .
- If the degree of the numerator is greater than the degree of the denominator, then there is no H.A. However, there is a slant or other type asymptote, found by the quotient of the division of the rational function.
Examples:

7. Solving Rational Equations
Extraneous Solutions – solutions found by algebraic processes that are not actually zeroes
1. Find the Lowest Common Denominator (must contain factors of the smallest
multiplicity of each factor)
2. Multiply each term by the LCD
3. Solve using inverse operations
4. Check for extraneous solutions
Examples:

8. Sign Analysis When 𝑓 𝑥 <≤𝑜𝑟≥>0 (interval notation answers!)
Start by factoring! Then identify it as a zero, point discontinuity, or vertical asymptote.
discontinuity
zero
zero
(+) (−)(−)
𝑓 𝑥 =
positive
value
value
value
Examples: