1. 5.7 Graphs of Rational Functions

2. Rational Function Rational function: Ex 1) Sketch x y x y
x-axis and y-axis are Asymptotes!
Rational Function
Rational function:
Doesn’t touch y-axis
Ex 1) Sketch x y x y
As x ↑ , y gets closer to 0 1 1 -1 -1 2 -2 -½ 3
1/3 -3
-1/3
As x ↓ , y ↑ 4 -4 -¼ 2 -½ -2
Doesn’t touch x-axis 4 -¼ -4

3. Vertical & Horizontal Asymptotes
Vertical Asymptotes:
when denom = 0
Horizontal Asymptotes:
If n < m, y = 0
If n = m, y = a/b
If n > m, no H.A.
num exp < denom exp
exponents of leading coeff.
num exp = denom exp
num exp > denom exp

4. x0 Ex 2) Sketch the graph of V.A.: H.A.: Shift right 1 Stretch (x2) x
Numer exp < denom exp
→ y = 0
Shift right 1
Stretch (x2) x y 2 2 3 1 5 -2 -1 -1 -3 -½

5. Oblique Asymptotes Case 3) n > m:
If expon on numer is exactly 1 more than expon on denom → oblique asymptote
O.A.:

6. Ex 3) Find all asymptotes
V.A.:
H.A.:
none
O.A.:
O.A.: ↓

7. T.O.O. Ex 3 cont…. (Find all asymptotes)
V.A.:
H.A.:
V.A.:
H.A.:

8. Ex 4) Sketch the graph of x y V.A.: H.A.: 1 -1 .5 -4/3 1.5 -4/3 3 1/3
.125
2.25
1.8 -1
1/3 -2
.125
-.5 .8

9. Homework
#511 Pg – 19 odd