Understanding Properties of Rational Functions

Section 5.2 – Properties of Rational Functions
 
Defn:
 
  Rational Function
 
The functions 
p
 and 
q
 are polynomials.
 
The domain of a rational function is the set of all real
numbers except those values that make the
denominator, q(x), equal to zero.
Section 5.2 – Properties of Rational Functions
 
  Domain of a Rational Function
 
or
 
(-, -4)  (-4, )
 
{
x 
| 
x
 
 –4}
 
graph
Section 5.2 – Properties of Rational Functions
  Domain of a Rational Function
 
or
 
(-, 2)  (2, )
 
{
x 
| 
x
 
 2}
 
graph
Section 5.2 – Properties of Rational Functions
  Domain of a Rational Function
 
or
 
(-, -3)  (-3, 3)  (3, )
 
{
x 
| 
x
 
 –3, 3}
 
graph
Section 5.2 – Properties of Rational Functions
  Domain of a Rational Function
 
or
 
(-, -3)  (-3, 5)  (5, )
 
{
x 
| 
x
 
 –3, 5}
 
Linear Asymptotes (vertical, horizontal, or oblique)
 
Lines in which a graph of a function will approach. By approach we mean each
successive value of X puts the graph closer to the asymptote than the previous
value.
 
Vertical Asymptote
 
A vertical asymptote exists for any value of x that makes the
denominator zero 
AND
 is not a value that makes the numerator
zero, in this case the factors would cancel.
 
Example
Section 5.2 – Properties of Rational Functions
 
A vertical asymptotes exists at x = -5.
 
graph
Asymptotes
 
Vertical Asymptote
 
Example
Section 5.2 – Properties of Rational Functions
 
A vertical asymptote does not exist at x = 3 as it is a value
that also makes the numerator zero.
 
A hole exists in the graph at x = 3.
 
graph
 
Horizontal Asymptote
 
A horizontal asymptote exists if the largest exponents in
the numerator and the denominator are equal,
Section 5.2 – Properties of Rational Functions
Asymptotes
 
If the largest exponent in the denominator is equal to the
largest exponent in the numerator, then the horizontal
asymptote is equal to the ratio of the coefficients.
 
or
 
if the largest exponent in the denominator is larger than
the largest exponent in the numerator.
Asymptotes
 
Horizontal Asymptote
 
Example
Section 5.2 – Properties of Rational Functions
 
A horizontal asymptote exists at y = 0.
 
A horizontal asymptote exists at y = 5/2.
 
graph
 
Oblique (slant) Asymptote
 
An oblique asymptote exists if the largest exponent in the
numerator is one degree larger than the largest exponent
in the denominator.
Section 5.2 – Properties of Rational Functions
Asymptotes
 
Other non-linear asymptotes can exist for a rational
function.
 
**Note**
Asymptotes
 
Oblique Asymptote
 
Example
Section 5.2 – Properties of Rational Functions
 
An oblique asymptote exists.
 
Long division is required.
We ignore the remainder if
it exists
 
graph
Asymptotes
 
Oblique Asymptote
 
Example
Section 5.2 – Properties of Rational Functions
 
An oblique asymptote exists.
 
An oblique asymptote
exists at y = 2x
 
Long division is required.
 
graph
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Rational functions are expressed as the ratio of two polynomials. The domain of a rational function excludes values that make the denominator zero. Various examples illustrate how to determine the domain and identify asymptotes in rational functions. Vertical asymptotes exist where the denominator is zero but not the numerator. This content provides insights into the properties of rational functions and how to interpret them graphically.


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  1. Section 5.2 Properties of Rational Functions Defn: Rational Function A function in the form: ? ? =?(?) ?(?) The functions p and q are polynomials. The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero. ? ? =?2 1 ? ? =2?2 4 2 ? = ?2 4 ? 1 ? + 5

  2. Section 5.2 Properties of Rational Functions Domain of a Rational Function ? ? =?2 3 ? + 4 ? + 4 = 0 ? = 4 ??????: {x | x 4} or (- , -4) (-4, )

  3. graph

  4. Section 5.2 Properties of Rational Functions Domain of a Rational Function ? ? =?2 4 ? 2 ? 2 = 0 ? = 2 ??????: {x | x 2} or (- , 2) (2, )

  5. graph

  6. Section 5.2 Properties of Rational Functions Domain of a Rational Function 2 ? = ?2 9 ?2 9 = 0 (? 3)(? + 3) = 0 x = 3,3 ??????: {x | x 3, 3} or (- , -3) (-3, 3) (3, )

  7. graph

  8. Section 5.2 Properties of Rational Functions Domain of a Rational Function 2? + 3 ?2 2? 15 ? = ?2 2? 15 = 0 (? 5)(? + 3) = 0 x = 3,5 ??????: {x | x 3, 5} or (- , -3) (-3, 5) (5, )

  9. Section 5.2 Properties of Rational Functions Linear Asymptotes (vertical, horizontal, or oblique) Lines in which a graph of a function will approach. By approach we mean each successive value of X puts the graph closer to the asymptote than the previous value. Vertical Asymptote A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero, in this case the factors would cancel. Example =(? 4)(? + 4) ? + 5 ? ? =?2 16 x = 5 ? + 5 VA: ? = 5 A vertical asymptotes exists at x = -5.

  10. graph

  11. Section 5.2 Properties of Rational Functions Asymptotes Vertical Asymptote Example =(? + 2)(? 3) (? 4)(? 3) ?2 ? 6 ?2 7? + 12 x = 3,4 ? ? = A vertical asymptote exists at x = 4. VA: ? = 4 A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero. A hole exists in the graph at x = 3.

  12. graph

  13. Section 5.2 Properties of Rational Functions Asymptotes Horizontal Asymptote A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, or if the largest exponent in the denominator is larger than the largest exponent in the numerator. If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients. If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is ? = 0.

  14. Section 5.2 Properties of Rational Functions Asymptotes Horizontal Asymptote Example ? ? =5?3 2?2 7 2?3 7? + 10 HA: ? =5 A horizontal asymptote exists at y = 5/2. 2 ? 6 ? ? = ?2 7? + 12 HA: ? = 0 A horizontal asymptote exists at y = 0.

  15. graph

  16. Section 5.2 Properties of Rational Functions Asymptotes Oblique (slant) Asymptote An oblique asymptote exists if the largest exponent in the numerator is one degree larger than the largest exponent in the denominator. **Note** Other non-linear asymptotes can exist for a rational function.

  17. Section 5.2 Properties of Rational Functions Asymptotes Oblique Asymptote Example An oblique asymptote exists. Long division is required. We ignore the remainder if it exists ? ? =?2+ 1 ? ? + x + 2 0 1 x x ?2 0 0? An oblique asymptote exists at y = x. OA: ? = ?

  18. graph

  19. Section 5.2 Properties of Rational Functions Asymptotes Oblique Asymptote Example ? ? =4?4+ 2?2+ ? 1 2?3+ 3? An oblique asymptote exists. Long division is required. 2? + + + + + + 4?2 + + 3 4 3 2 2 x 3 x 4 4?4 x 0 x 2 6?2 x x 1 An oblique asymptote exists at y = 2x OA: ? = 2?

  20. graph

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