MATH 1314 College Algebra

1. MATH 1314 College Algebra Properties & Graphs of Rational FunctionsSection: ____

2. Rational Functions 1 • Form 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.

3. 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 graphof intersects HA at that -value.

4. Rational Functions 3 • Continued: • Graph may have Oblique Asymptote(OA). • ifdegree 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 factorreduces 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.

5. Rational Functions 4 • Example: • Domain: • Range: • VA: • HA: • OA: • Intercepts:

6. Rational Functions 5 factored reduced • 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. y-int. x-int. zeros of numerator in reduced.y-int.

7. Rational Functions 6 factored, can’t reduce • 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. y-int. x-int. zeros of numerator in reduced.y-int.

8. Polynomial & Rational InequalitiesSection: ____

9. 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?”

10. Poly. & Rational Inequalities 2 • Example: 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: r r 0 5 Interval I Interval II Interval III r r

11. Poly. & Rational Inequalities 3 • Example: Solve . • Rewrite inequality in standard form. Find zeros: • real zero fromdenominator.(VA excluded) • Test intervals with . • Interval I : Test • Interval II: Test • Solution: Use LCD r r 3 Interval I Interval II

12. Poly. & Rational Inequalities 3 For which -values is graph below -axis()?Intervals: use ) or ( along -axis. • Example: Use the graph of to solve the inequality. For which -values is graph at or above -axis()?Intervals: use ] or [ at zeros, but ) or ( everywhere else along -axis.