Class notes AP Precalculus

Worksheet A: (Topic 1.7) Rational Functions and End Behavior Name: Solutions
Directions: For each of the following, determine if the given rational function has a horizontal asymptote. If it does,
write the equation of the horizontal asymptote.

3x 2 − 1 x3 + 2 x 2 + x + 4 5 x3 − 2 x 2 − 1
1. f ( x ) = 2 2. g ( x ) = 3. h ( x ) =
2 x + 5x + 7 5x2 + 7 x + 8 x4 − 6

Horizontal Asymptote: Y or N Horizontal Asymptote: Y or N Horizontal Asymptote: Y or N
3 If Yes, Equation: __________ If Yes, Equation: 𝑦 = 0
If Yes, Equation: 𝑦 = 2



6 x3 + 2 x + 3
5. r ( x ) =
( 2 x − 1)( 5 x + 6 ) (x − 3)
2
4. k ( x ) =
2

2 x 2 − 11x + 4 ( x + 3)( x − 6 ) 6. q ( x ) =
3x3 + 4 x 2 + 7
Horizontal Asymptote: Y or N Horizontal Asymptote: Y or N Horizontal Asymptote: Y or N
If Yes, Equation: __________ If Yes, Equation: 𝑦 = 10 If Yes, Equation: __________




( 3x − 1)
2
8. y =
( x − 2 )( 4 − x ) ( 2x 2
+ 3) ( x − 4 )
2

7. p ( x ) = s ( x) =
( x + 3) 9.
(x + 5) ( x − 2 )
2
2 x 2 + 3x + 5 2


Horizontal Asymptote: Y or N Horizontal Asymptote: Y or N
Horizontal Asymptote: Y or N
9
If Yes, Equation: 𝑦 = If Yes, Equation: 𝑦 = −1
2 If Yes, Equation: __________




Directions: Write limit statements for the end behavior of the following rational functions.
2 x3 − 5 x + 6 ( 4 x + 3) x2
2
10. y = 3 11. y= 12. y =
6 x + 10 x 2 − 4 x − 12 ( 3x − 1)( 2 x + 5) ( x − 1)
3



1
Left: lim 𝑦 = 8 Left: lim 𝑦 = 0
𝑥→−∞ 3 Left: lim 𝑦 = 𝑥→−∞
𝑥→−∞ 3
1 8 Right: lim 𝑦 = 0
Right: lim 𝑦 = Right: lim 𝑦 = 𝑥→∞
𝑥→∞ 3 𝑥→∞ 3




𝐖𝐨𝐫𝐤𝐬𝐡𝐞𝐞𝐭 𝐀: Topic 1.7 Rational Functions and End Behavior