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MATH 1314 College Algebra
Properties & Graphs of Rational Functions
Section: ____
Rational Functions 1Form of Rational(fraction) function:
and are polynomial functions, but .
Domain: all real numbers , except those that make . To find domain of rational function ,
use number line method discussed in Section 3.1.
Rational function graphs are discontinuous(split apart). They split apart along vertical boundary lines called Vertical Asymptotes(VA). To find VA’s : must be in reduced form Set denominator , solve for .
Graph of cannot intersect VA’s.
Rational Functions 2 continued…
Graph may have a Horizontal Asymptote(HA). if degree of degree of ,
then HA at
OR if degree of degree of ,
then HA at
Graph may possibly intersect HA.To find -value where this may happen:
Solve equation above. If equation can be solved for , then graph
of intersects HA at that -value.
Rational Functions 3Continued:
Graph may have Oblique Asymptote(OA). if degree of degree of ,
then OA at To find equation of OA:
Divide by . Use Long Division. will be equation of OA.
Holes in graph(discontinuity): If a common factor reduces from ,
then is lost and hole occurs in graph at . Hole will be at using reduced function .
Intercepts: Find x and y-intercepts like before.
Rational Functions 4Example:
Domain:
Range:
VA:
HA:
OA:
Intercepts:
(−∞ ,4)∪(4 ,∞)
(−∞ ,−3)∪ (−3 ,∞)
𝑥=4𝑦=−3
𝑛𝑜𝑛𝑒(0,0)
−∞ ∞
−∞∞
Rational Functions 5 Example:
Domain:
Equations of VA’s:
Equations of HA or OA:
Intercepts:
?Factor everything. What can x not equal in denominator?
? must be reduced. What still makes denominator zero?
?Check degree of numerator and denominator. Degrees are equal. Need ratio of leading coefficients.
?x-int. zeros of numerator in reduced .y-int.
𝑓 (𝑥 )=2 𝑥2+3𝑥−5𝑥2+𝑥−2
x-int. y-int.
factored
¿2𝑥+5𝑥+2
reduced
Rational Functions 6 Example:
Domain:
Equations of VA’s:
Equations of HA or OA:
Intercepts:
?Factor everything. What can not equal in denominator?
? must be reduced. What still makes denominator zero?
?Check degrees. Degree of numerator larger. Need quotient from dividing numerator by denominator. Use Long Division.
?x-int. zeros of numerator in reduced .y-int.
𝑓 (𝑥 )=𝑥3+27𝑥2−4
x-int. y-int.
factored, can’t reduce
Polynomial & Rational Inequalities
Section: ____
Poly. & Rational Inequalities 1 When solving Polynomial & Rational Inequalities, first be sure
to rewrite inequality in standard form so that zero is on the right side and simplified expression is on the left side, such as: , , , To solve Polynomial or Rational Inequalities(standard form):
Polynomial: Find real zeros & label on number line.Rational: Find real zeros of numerator & denominator & labelon number line. Denominator zeros(VA’s) never use ] or [.
Test a number from each interval. Use inequality in standard form. Choose interval that satisfies inequality in standard form.
Remember: and not inclusive. Use or at zeros. and inclusive. Use ] or [ at zeros, except at VA’s.
If using graph of : means “for which -values is graph below -axis?” means “for which -values is graph above -axis?” means “for which -values is graph on and below -axis?” means “for which -values is graph on and above -axis?”
Poly. & Rational Inequalities 2Example: Solve , where
Rewrite inequality in standard form. Find zeros: , , , are real zeros.
A bracket is used at zeros for or .
Test intervals using . Interval I : Test
Interval II : Test
Interval III : Test
Solution:
−∞ ∞0 5
Interval I Interval II Interval III
r
r
¿ r r
−1 1 6
¿
Poly. & Rational Inequalities 3Example: Solve .
Rewrite inequality in standard form. Find zeros:
real zero fromdenominator.(VA excluded)
Test intervals with . Interval I : Test
Interval II : Test
Solution:
−∞ ∞3
Interval I Interval II
3(2 )−3
>0 r
r
()
Use LCD 𝑥
𝑥−3−
(𝑥−3)𝑥−3
>03
𝑥−3>0
24
3(4 )−3
>0
−3>0
3>0
(3 ,∞ )
∞−∞
Poly. & Rational Inequalities 3Example: Use the graph of to solve the inequality.
For which -values is graph below -axis()?Intervals: use ) or ( along -axis.
𝑓 (𝑥)<0
()()
(−4,0)∪(6 ,∞)
𝑓 (𝑥)≥0
()¿
(−∞ ,−4 )∪ ¿For which -values is graph at or above -axis()?Intervals: use ] or [ at zeros, but ) or ( everywhere else along -axis.