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.