3.5 Domain of a Rational FunctionThurs Oct 2

Do NowFind the domain of each function

1)

2)

Ch 1 Test Review

• Retakes: If you plan on retaking this test for 90%, see me at end of class

• Retakes must be scheduled for this week

Rational Function

• A rational function is a function f that is a quotient of two polynomials

where q(x) is not the zero polynomial

The domain of f(x) consists of all inputs x for which q(x) is not 0

Graphs of Rational Functions

• Various examples of graphs of rational functions can be found on page 301

Finding the domain

• To find the domain of a rational function, set the denominator equal to 0, and solve for x

• Note: any factors that you could cancel out still count towards the domain!

Ex

• Find the domain of each• 1) • 2) • 3) • 4) • 5) • 6)

Asymptotes

• An asymptote is a line that the function’s graph gets very close to but may not cross

• There are 3 types of asymptotes– Vertical asymptotes (x = )– Horizontal asymptotes (y = )– Oblique asymptotes (y = mx + b)

Vertical Asymptotes

• The line x = a is a vertical asymptote of the rational function p(x)/q(x) if:

– X = a is a zero of the denominator– P(x) and q(x) have no common factors

Ex

• Determine the vertical asymptotes for the graph of

You try

• Find all vertical asymptotes for each function• 1)

• 2)

• 3)

Closure

• Find the vertical asymptotes for

• HW: p.316 #7-13 odds, 69

3.5 Horizontal and Oblique AsymptotesMon Oct 6

• Do Now• Find the vertical asymptotes of

HW Review: p.316 #7-13 69

Horizontal Asymptotes

• The line y = b is considered a horizontal asymptote of p(x)/q(x) if:– As x approaches infinity, y approaches b– As x approaches neg. infinity, y approaches b

Horizontal asymptotes only refer to a graph’s end behavior

- A graph can cross horizontal asymptotes in the middle of the graph

Horizontal Asymptotes

• 3 cases: For each case you want to consider the highest power in the numerator and denominator– Case 1: Denominator’s power greater: y = 0– Case 2: Numerator’s power greater: none– Case 3: Powers are equal: y = a/b where a and b

are the lead coefficients of the num and denom

Ex 1

• Find the horizontal asymptote

Ex 2

• Find the horizontal asymptote of

Notes

• The graph of a rational function never crosses a vertical asymptote

• The graph of a rational function might cross a horizontal asymptote

Oblique Asymptotes

• A function has an oblique asymptote if the numerator’s power is exactly one higher than the denominator’s power

• To determine oblique asymptotes, we use long division

• Graphs can cross oblique asymptotes

Ex

• Find all asymptotes of

Ex

• Find all asymptotes of the function

Closure

• What is the difference between a horizontal and oblique asymptote? How do you find each one?

• HW: p.316 #1 3 5 15-25 odds

3.5 Graphing Rational FunctionsTues Oct 7

• Do Now• Find all asymptotes• 1)

• 2)

HW Review: p.316 #1-5 15-25

Graphing Rational Functions

• 1) Find all asymptotes– Remember, can’t have both oblique and horizontal

asymptotes• 2) Find x and y intercepts– Plug in 0 for y and solve, for x and solve

• 3) For each region, test x-coordinates to determine where each curve occurs

Ex

• Graph

Ex2

• Graph

Ex3

• Graph

Closure

• Graph

• HW: p.317 #29-57 odds