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2.6 Rational Functions and Asymptotes
Transcript of 2.6 Rational Functions and Asymptotes
Maggie Nelson 2.6 Rational Functions Rational Functions The vertical asymptote= the zeros of D(x). Vertical Asymptotes First determine at highest degree of polynomials on top and bottom Horizontal Asymptotes A utility company burns coal to generate electricity. The cost C (in dollars) of removing p% of smoke stakes is given by Cost of Removing Pollutants 1. If highest degree of N(x) highest degree D(x) then asymptote is: N(x)
D(x) f(x)= ____ The domain of f(x) is always all real numbers except where D(x)=0 vertical asymptote:
x=2 D(x) N(x) The simplest rational function: ______ f(x)= f(x)= f(x)= 1 ___ x ** ** 1. If highest degree of N(x) highest degree of D(x)
then asymptote is: < 2. If highest degree of N(x) highest degree of D(x)
then asymptote is: = y = coefficient of highest degree of N(x) coefficient of highest degree of D(x) ____________________________________________ 3. If highest degree of N(x) highest degree of D(x)
then: y = 0 > remember to factor first... f(x)= 1 _______ x -x-6 2 f(x)= 1 __________ (x-3)(x+2) C= 80,000p __________ 100-p for 0<p<100. Graph this function. You want to require utility companies to remove 90% of the pollutants from the smoke stack emissions. It currently is only 85%. How much additional cost would the utility company incur as a result of the new law? y= 2x + 3x + x+2 __ Holes factor the numerator and denominator f(x)= x +x-2
x -x-6 2 2 _______ (x-1)(x+2)
(x+2)(x-3) __________ (x-1)
(x-3) _____ x+2=0 x=-2 set the canceled out pair equal to zero x-2 y = 0 < f(x)= x _______ x -5 2 horizontal asymptote:
y=0 ex. 2. If highest degree of N(x) highest degree
of D(x) then asymptote is: = y = coefficient of highest degree of N(x) coefficient of highest degree of D(x) ____________________________________________ ex. f(x)= 2x x -1 2 2 _____ horizontal asymptote: y=2 3. If highest degree of N(x) highest degree
of D(x) then: NO horizontal asymptote > ex. f(x)= x 3 ____ x 2 horizontal asymptote:
none **to find the ordered pair of the hole, plug in the x-value back into the simplified rational function** 3 2 _________________________ x +3x+15 2 vertical asymptote horizontal asymptote x=-2 x=3 1 _____ ex. NO horizontal asymptote continuous functions= no vertical asymptotes (D: all reals) & no holes Suggested Homework page 148
#11-16, 20-35 (mod 5) except 30, plus 37-45 Practice Problems Find the domain, asymptotes, and holes (if there are any) ** ** 1. f(x)= 4x +3x+5 2 2x +x-6 2 _____________ 2. f(x)= x -4 x -3x-10 _____________ 2 2 3. f(x)= x +3x+2 x-1 2 _____________ Domain:
H. Asymptote: all reals, x≠ -2, (3/2) Domain:
H. Asymptote: x=-2, (3/2) none y=2 all reals, x≠ 5, -2 (-2,0) x=5 y=1 all reals, x≠1 none x=1 none Domain:
H. Asymptote: Online Help Khan Academy Video: http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/asymptotes-of-rational-functions Hole: (-2, 3/5) C= 80,000(85) ___________ 100-85 = C= 80,000(90) ___________ 100-90 = 720,000-453,333.33 = $266,666.67 $453,333.33 $720,000 $266,666.67