1. RationalExpressions

2. Rational functions and their graphs • A rational function can be written as f(x)=where P(x) and Q(x) are polynomial functions and Q(x) can not equal zero. • Example:

3. Vertical and horizontal asymptopes • To find vertical asymptotes, set the denominator equal to zero. • A graph ahs at most one horizontal asymptote if: (the m degree of numerator and n is degree of denominator) • m<n, then y=0 is the horizontal asymptote • m=n, then line y= • m>n, then there is no horizontal asymptote

4. Rational expressions • A rational expression is in simplest form when its numerator and denominator are polynomials that have no common divisors. • To get an expression in simplest form, divide out the common factors. • Ex: • Factor: • Divide out common factors to get the answer • Set the common factors equal to zero to get restrictions. Restrictions are at x=-4 or -5

5. Multiplying • When multiplying, use what you know about simplifying rational expressions to work them. Multiply across. • Ex: x • Ex: X

6. Dividing rationals When dividing, use the term “copy, switch, flip”. Copy the first term of the expression, switch the dividing sign to a multiplication sign, and flip the last expression to its reciprocal. • Ex: / • Ex: /

7. Finding least common multiples • To add or subtract, you have to find the LCD. To do this, find the least common multiple. • Ex: Find the least common multiple of each pair of polynomials. and . (x+1)(x-1) (x+1)(x+1) Both pairs have a (x+1) in common. To find LCD, multiply each by what they’re missing. Multiply the first pair by the other (x+1) that it’s missing, and multiply the second pair by the (x-1) that it’s missing. The LCD would end up being (x+1)(x+1)(x-1).

8. Adding and subtracting expressions • Find the LCD of the expressions to add and subtract. Add across the numerators and simplify. • Ex: Factor the denominators and find the LCD Multiply numerator and denominator by what each expression is missing in the LCD

9. Adding and subtracting ex: • Ex: + • Ex: • Ex: • Ex: -

10. Complex fractions • A Complex Fraction is a fraction that has a fraction in its numerator or denominator. Its can also be found in both numerator and denominator. • Examples: • To simplify a complex fraction such as you can multiply the numerator and denominator by their LCD bd. Or you can divide the numerator by the denominator .

11. Simplifying complex fractions Ex: • First find the LCD of all the rational expressions. = • The LCD is xy. Multiply the numerator and denominator by xy. = • Use the distributive property = • Simplify

12. Try it yourself…

13. Solving rational equations • Solve and then check each solution. = Write cross products… 5(x^2-1) = 5(2x-2) Distributive Property… 5x^2-5 = 30x – 30 Write in standard form…5x^2 – 30x +25 = 0 Divide each side by 5… x^2 – 6x +5 = 0 Factor… (x-1)(x-5) = 0 Zero-Product Property… x=1 or x=5

14. Try it yourself… = =

15. Unit 3Be ready to go when the bell rings! Example 1:

16. Unit 3 Example 2:

17. Unit 3 Example 3:

18. Unit 3 Example 4:

19. Unit 3 Example 5:

20. Unit 3 Example 6: xint: VA: HA: Hole:

21. Unit 3 Example 7: Describe the translation, list both asymptotes.

22. MSL (Common Exam)

23. MSL (Common Exam)