Rational
Rational Functions
Functions
• How do we graph rational functions?
• How do we transform rational functions
by changing parameters?
HoltMcDougal
Algebra 2Algebra 2
Holt
Rational Functions
A discontinuous function is a function
whose graph has one or more gaps or breaks.
The hyperbola graphed in Example 2 and
many other rational functions are
discontinuous functions.
A continuous function is a function whose
graph has no gaps or breaks. The functions
you have studied before this, including linear,
quadratic, polynomial, exponential, and
logarithmic functions, are continuous
functions.
Holt McDougal Algebra 2
Rational Functions
Holt McDougal Algebra 2
Rational Functions
Some rational functions, including those
whose graphs are hyperbolas, have a
horizontal asymptote. The existence and
location of a horizontal asymptote depends
on the degrees of the polynomials that make
up the rational function.
Note that the graph of a rational function can
sometimes cross a horizontal asymptote.
However, the graph will approach the
asymptote when |x| is large.
Holt McDougal Algebra 2
Rational Functions
Holt McDougal Algebra 2
Rational Functions
Graph the function (p < q). State the domain and range.
x
1. y  2
x 1
 x  1 x  1 
VA: x   1 , x  1
HA:
y0
x
y
x-intercepts:
Domain:
a ll R x   1 , 1
Range:
a ll R y  0
x
y
 3  3 / 8  .5 2 / 3
 2  2 / 3 .5  2 / 3
2 2/3
3 3/8
Holt McDougal Algebra 2
x0
Rational Functions
Graph the function (p < q). State the domain and range.
2x
2. y  2
x 4
zeros:
Domain:
VA: None
HA: y  0
x
y
 3  6 /13
 2  4/8
1  2 / 5
1 2/5
Holt McDougal Algebra 2
all R
Range:
a ll R y  0
x
y
2 4/8
3 6 /13
x0
Rational Functions
Graph the function (p > q). State the domain and range.
x
3. y 
x 1
2
zeros:
Domain:
VA: x  1
HA: None
all R x   1
Range:
Diagonal asymptotes:
x
3
2
1
2
y
9/ 2
4
 1/ 2
 4/3
Holt McDougal Algebra 2
y  0, y  4
x
y
 1.5 4.5
0.5  .17
x0
Rational Functions
Graph the function (p > q). State the domain and range.
x 1
4. y 
x2
2
zeros: N o ne
Domain:
VA: x  2
HA: None
all R x  2
Range:
Diagonal asymptotes:
x
y
0  1/ 2
12
3 10
4 17 / 2
Holt McDougal Algebra 2
x
y  0, y  8
y
1  2 /3
5 26 / 3
Every y - line worth 2
Rational Functions
Graph the function (p > q). State the domain and range.
x 2 6  x  3 
x  3x  18
5. y 
x
VA: x  0
HA: None
zeros: x
 6, x  3
Domain:
all R x  0
Range:
Diagonal asymptotes: all R
x
4
2
2
4
y
7/2
10
4
5/2
Holt McDougal Algebra 2
x
y
 8  11 / 4
6 6
Every line worth 2
Rational Functions
Graph the function (p > q). State the domain and range.
x  32 x  3 
x 9
6. y 
2x
zeros: x
Domain:
VA: x  0
HA: None
all R x  0
Range:
Diagonal asymptotes: all R
x
y
 2 5/4
1 4
1 4
2  5/4
Holt McDougal Algebra 2
x
y
 4  7 /8
4 7 /8
  3, x  3
Rational Functions
Graph the function (p = q). State the domain and range.
x 1
7. y 
x3
VA:
HA:
x3
y 1
x
1
2
4
5
y
1
3
5
3
Holt McDougal Algebra 2
zeros: x
Domain:
all R x  3
Range:
all R y  1
 1
Rational Functions
Graph the function (p = q). State the domain and range.
2
2x
8. y  2
x 1
 x  1 x  1 
VA: x   1 ,
HA: y  2
x
3
2
2
3
Domain:
x1
a ll R x   1 , 1
Range:
all R y  2
x
y
y
18 / 8
8/3
8/3
18 / 8
zeros:
 .5  2 / 3
.5  2 / 3
Holt McDougal Algebra 2
x0
Rational Functions
Lesson 6.4 Practice C
Holt McDougal Algebra 2