3.5 domain of a rational function thurs oct 2 do now find the domain of each function 1) 2)
TRANSCRIPT
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