Download Ch. 3.6

Class Work
1. Find the real zeros by factoring.
P(x) = x4 – 2x3 – 8x + 16
2. Divide.
4 x3  2 x 2  2 x  3
2x 1
3. Find all the zeros of the polynomial.
P(x) = x3 – 2x2 + 2x – 1
Sec 3.6 Rational Functions
Objectives:
•To understand how to find holes,
vertical, horizontal and slant
asymptotes.
Vertical Asymptotes
The line x = a is a vertical asymptote if y
approaches ± as x approaches a from the left
or right.
Vertical Asymptotes - A rational function has a
vertical asymptote at x = c when
• c is a zero of the denominator
• c is NOT a zero of the numerator.
Ex 1. Find the vertical asymptotes.
2
a)
x5
6
b)
2x  9
Holes
When a number c is both a zero of the
numerator and the denominator then there is
a hole at x = c.
Ex 2. Find the holes of the function.
x2  4
a)
x2
x 3  64
b)
x4
Horizontal Asymptotes
Let
ax n  ...
f ( x)  m
cx  ...
• If n < m, then there is a horizontal asymptote at y = 0.
• If n = m, then there is a horizontal asymptote at
a
y
c
where a and c are the leading coefficients of the
numerator and the denominator.
• If n >m, then there is no horizontal asymptote,
but it does have a slant asymptote.
Ex 3. Find all holes and vertical and
horizontal asymptotes.
a)
x2  x  2
f ( x)  3
x  5x2  6 x
x 2  3x  4
b) f ( x) 
2 x2  4 x
3x 2  12 x
c) f ( x)  3 2
x  x  20 x
Slant Asymptotes
Remember, slant asymptotes occur when the
degree of the numerator is greater than the
degree of the denominator.
To find slant asymptotes, divide the numerator
by the denominator and the quotient will give
you the equation of the slant asymptote.
Ex 4. Find all asymptotes and holes.
a)
x2  4x  5
f ( x) 
x 3
2
x
b) f ( x)   x  6
x2
This is the graph of Ex 4 part a.
Class Work
Find all asymptotes and holes.
2
x
4. f ( x)   3x  4
2 x2  4 x
5.
x  2 x  5x  6
f ( x) 

2
x  3x  2
6.
x3  x 2  4 x  4
f ( x) 
x2  x  6
3
2
 x  2   x 2  4 x  3
x 2  3x  2
HW 3.6 Worksheet