DrDelMath

SUMMARY
Chapter 4:  Rational Functions and Conics

Section 4.1: Rational Functions and Asymptotes

Definition:  A rational function is a function which may be written as the quotient of two polynomial functions.
Symbolically we can say that a rational function can be written in the form where N and D are polynomial functions and D is not zero.
The rule of a rational function can be written in the form where N(x) is equal to a polynomial and D(x) is equal to a polynomial.

Convention: As with all functions the convention for the domain of a rational function is all real numbers for which the rule is defined ( makes sense). Therefore the domain of a rational function is all real numbers except the zeros of the denominator.

Fact: The domain of a rational function is all real numbers except the zeros of the denominator.

Definition: The vertical line whose equation is x = a, is a vertical asymptote of the graph of a function f if any one of the following are true:

• f(x) increases without bound as x approaches a from the left
• f(x) increases without bound as x approaches a from the right
• f(x) decreases without bound as x approaches a from the left
• f(x) decreases without bound as x approaches a from the right

Fact: Vertical asymptotes of a rational function occur at the real zeros of the denominator which are not zeros of the numerator.

Procedure: To find the vertical asymptotes of a rational function find the real zeros of the denominator and eliminate those which are also zeros of the numerator.

Definition: The horizontal line whose equation is y = b, is a horizontal asymptote of the graph of a function f if any one of the following are true:

• f(x) approaches b as x increases without bound
• f(x) approaches b as x decreases without bound

Procedure: To find a horizontal asymptote of a rational function :
    1.   If m < n, the line y = 0 is a horizontal asymptote.
    2.   If m = n, the line .
    3.   If m > n, there is no horizontal asymptote.

Procedure: To find a horizontal asymptote of a rational function:
    1.  Dividing each term of the numerator and each term of the denominator by the highest power of x in the denominator
    2.  Determine the behavior as x increases (or decreases) without bound by observing which terms approach zero

Fact: The graph of a function f cannot intersect any of its vertical asymptotes.
Fact: The graph of a function f may intersect its horizontal asymptote.
Fact: The graph of a function f need not intersect its horizontal asymptote but it may intersect its horizontal asymptote any number of times, including infinitely many times.

Fact: If a zero of the denominator of a rational function is also a zero of the numerator, then there is no vertical asymptote at that point. Rather there is a "hole" in the graph, that is just one point is missing in the graph.

Section 4.2: Graphs of Rational Functions

Procedure: The zeros of a rational function f are found by solving the equation resulting from f(x) = 0. Therefore we find the zeros of a rational function by solving the equation .In view of the Zero Quotient Property we have the following fact.

Fact: The zeros of a rational function are the zeros of its numerator which are not also zeros of the denominator.

Procedure: To sketch the graph of a rational function:

1. Find the zeros of the numerator, note which are real numbers
2. Plot the real zeros and label them, because they are x-intercepts of the function
3. Find the zeros of the denominator, note which are real numbers and note which are not also zeros of the numerator
4. Plot the real zeros of the denominator and erect vertical lines through those which are not zeros of the numerator. These are the vertical asymptotes.
5. Erect vertical lines through each of the x-intercepts.
6. Note that the vertical lines through the x-intercepts and the vertical asymptotes divide the plane into several strips
7. Pick a convenient number in each strip and calculate its corresponding range element
   1. If this range element is positive, exclude the lower half of the strip-the part below the x-axis
   2. If this range element is negative, exclude the upper half of the strip-the part above the x-axis
8. Determine the horizontal asymptote, if any, by
   1. Dividing each term of the numerator and each term of the denominator by the highest power of x in the denominator
   2. Determine the behavior as x increases (or decreases) without bound by observing which terms approach zero
9. Sketch the graph by drawing only in the regions which have not been excluded. Be sure to draw through the x-intercepts and be sure to indicate all asymptotic behavior.

Oblique Asymptotes: If f is a rational function and the degree of N is 1 more than the degree of D, then f can be expressed in the form where the degree of R is less than the degree of D. In this case the line y = mx + b is an oblique asymptote for the graph of f.

Section 4.3: Partial Fractions

Section 4.4: Conics

Section 4.5: Translations of Conics