Transcript QUADRATIC FUNCTIONS

SECTION 3.2
RATIONAL FUNCTIONS
RATIONAL FUNCTIONS
Rational functions take on the
form:
p(x)
f(x) =
q(x)
Where p(x) and q(x) are
polynomials and q(x)  0
EXAMPLES:
3
2x - 3
f(x) =
x+1
x + 3x - 1
g(x) =
2
x +1
2
x - x - 6
h(x) =
2
x - 4
DOMAIN OF RATIONAL
FUNCTIONS
ANY VALUE WHICH ZEROS OUT
THE DENOMINATOR MUST BE
EXCLUDED FROM THE DOMAIN.
Do Example 1
GRAPHS OF RATIONAL
FUNCTIONS
ASYMPTOTE - a line which the
graph will approach but will never
reach.
HORIZONTAL
ASYMPTOTES
When the degree in the
numerator is equal to the
degree in the denominator.
Example: Graph
2
3x - 4
f(x) =
2
x + 1
By studying the graph, we can
describe what is happening
using the following symbols:
As x   , f(x)  3
As x  -  , f(x)  3
We say f(x) has a horizontal
asymptote of y = 3.
To see the algebraic reasoning
behind this, divide the rational
expression through by x2 and
examine what happens as x gets
huge.
2
3x
4
- 2
2
x
x
2
x
1
2 +
2
x
x
The numerator goes to 3 and the
denominator goes to 1.
EXAMPLE:
2
4x - 8x
Graph: f(x) =
on - 10  x  10
2
x + 1
4x 2
8x
- 2
2
x
x
2
x
1
2 +
x
x2
Again, algebraically, we can see that
the numerator goes to 4 and the
denominator goes to 1 as x gets huge.
Thus, this function has a horizontal
asymptote y = 4.
We can get even more specific with
our symbols when describing the
graph of the function and say the
following:
As x   , f(x)  4 As x  -  , f(x)  4+
QUESTION:
Can you find an easy way of
looking at the symbolic form
of a rational function in which
the degree in the numerator is
equal to the degree in the
denominator to find the
horizontal asymptote?
2
3x - 4
f(x) =
2
x + 1
2
4x - 8x
f(x) =
2
x + 1
Answer:
Just divide the leading
coefficient in the numerator by
the leading coefficient in the
denominator.
EXAMPLE:
Determine the equation of the
horizontal asymptote for the
graph of :
6
5
3x + 2x + 5
f(x) =
6
4
2x + x + 9
Horizontal Asymptote: y = 3/2
VERTICAL
ASYMPTOTES:
Vertical lines are always given
by an equation in the form x = c.
Here, the graph of the rational
function will approach a vertical
line, yet never quite reach it.
This means that this is a value x
will never equal.
Vertical asymptotes are nothing
more than domain restrictions, or
values for the variable that will
cause the denominator to equal 0.
EXAMPLE:
1
f(x) =
x - 2
x = 2 is not in the domain of f(x)
because replacing x with 2
would cause the denominator to
equal 0. Graph f(x).
Here, we can describe what is
happening to the graph of the
function by using the
following language:
As x  2 - , f(x) - 
As x 2+ , f(x) 
Thus, the vertical asymptote for
this function is x = 2.
Is this the only asymptote for this
function?
No! This function also has a
horizontal asymptote given by the
equation y = 0
As x gets huge, the numerator
goes to zero and the denominator
goes to 1.
1
x
x
2
x
x
In fact, any rational function in
which the degree in the
numerator is less than the
degree in the denominator will
have a horizontal asymptote of
y = 0 (or the x-axis).
EXAMPLE:
2
3x + 5
f(x) = 2
x - 4x + 4
We can either examine the graph to
determine the asymptotes, or we can
study the symbolic form, using the
tools we’ve learned.
The vertical asymptotes will be the
domain restrictions. What values will
zero out the denominator?
3x 2 + 5
f(x) =
2
(x - 2)
Thus, the only vertical asymptote is
x = 2.
2
3x + 5
f(x) = 2
x - 4x + 4
Now, for the horizontal
asymptotes.
The degree in the numerator is
equal to the degree in the
denominator.
Thus, the horizontal asymptote
is y = 3. Graph the function.
EXAMPLE:
Determine all asymptotes:
2
4x
f(x) =
x 2 + 2x - 8
2
4x
f(x) =
(x + 4)(x - 2)
H.A. y = 4
V.A.
x=-4
x= 2
SLANT ASYMPTOTES
When the degree in the
numerator is exactly one more
than the degree in the
denominator.
f(x) =
x
2
- 3x + 6
x - 2
f(x) =
x
2
- 3x + 6
x - 2
To determine the slant
asymptote, we simply do the
division implied by the
fraction line. We can use
synthetic division.
f(x) =
2
x
2
- 3x + 6
x - 2
1
-3
2
6
-2
1
-1
4
Thus, f(x) can be written as:
2
x - 3x + 6
4
f(x) =
= x - 1 +
x - 2
x - 2
x 2 - 3x + 6
4
f(x) =
= x - 1 +
x - 2
x - 2
As x gets huge, the fractional
part tends toward 0 and the
entire function tends toward
the linear function y = x - 1.
Graph the function.
EXAMPLE:
Determine all asymptotes of
the function below:
3
2
2x + 3x
f(x) = 2
x + x- 6
Vertical Asymptotes:
x=-3
x=2
Slant Asymptote:
2x + 1
2
x + x - 6
3
2
2x + 3x + 0x + 0
2x3 + 2x2 - 12x
x2 + 12x
x2 + x - 6
11x + 6
2x 3 + 3x 2
11x + 6
f(x) =
= 2x + 1 +
2
2
x + x- 6
x + x - 6
Slant Asymptote:
y = 2x + 1
Graph the function
CONCLUSION OF SECTION 3.2