1. Rational Functions By: Elena Fayda, Ana Maroto, Madelynn Walker

2. What is a rational function? *A rational function is when a polynomial is divided by another polynomial.

3. Steps for Solving

4. 1)Write each expression as a fraction 5x-25=1+1 5x-25=1+ 1 3x 3 3x 1 1

5. 2) Find the least common denominator (LCD) for the entire equation. 5x-25=1+1 5x-25=1+ 1 3x 3 3x 1 1 LCD: 3x

6. 3) Apply the LCD to each fraction within the problem. *You only have to multiply the fractions that do not already have the LCD 5x-25=1+1 5x-25=1+ 1 3x 3 3x 3 1 LCD: 3x 5x-25=1 (x)+1 (3x) 3x 3 (x) 1 (3x)

7. 4)Simplify each fraction but, DO NOT REDUCE! Just multiply the entire equation by the LCD (denominator) to cancel out the denominator 5x-25=1+1 5x-25=1+ 1 3x 3 3x 3 1 LCD: 3x 5x-25=1 (x)+1 (3x) 3x 3 (x) 1 (3x) 5x-25=1x+3x 3x 3x 3x ((9((9

8. 5)Rewrite the equation without the LCD in the denominator. 5x-25=1x+3x 3x 3x 3x 5x-25=x+3x (9

9. 5)Solve for the variable. 5x-25=1x+3x 3x 3x 3x 5x-25=x+3x 5x-25=4x -5x -25=-x -1 -1 x=25 ((9((9

10. Steps for Multiplying and Dividing

11. 1) Factor the equation. x+2 x+4 3x+12 x 2 -4 (x+2)(x+4) 3(x+4)(x+2)(x-2) (( 9

12. 2) Cross out the same factors on the top and bottom. x+2 x+4 3x+12 x 2 -4 (x+2)(x+4) 3(x+4)(x+2)(x-2) (x+2)(x+4) 3(x+4)(x+2)(x-2)

13. 3) Rewrite the equation with what is left. *Remember if there are only numbers left in the denominator, put a 1 in the numerator. x+2 x+4 3x+12 x 2 -4 (x+2)(x+4) 3(x+4)(x+2)(x-2) (x+2)(x+4) 3(x+4)(x+2)(x-2) 1 3(x-2)

14. *If you are dividing, then you need to do the same steps as above but change the sign to multiplication and switch the second fraction. (Keep it, Switch it, Flip it) 5x 6 10x 2 x 2 y y (5x 4 ) (y) (y) (10x 2 ) x 2 2

15. Steps for Adding and Subtracting

16. 1) 1) Find a common denominator leo 3x-4 + 2x-5 x+3 x+3 LCD: x+3

17. 2) Write expression by adding or subtracting the numbers leo 3x-4 + 2x-5 x+3 x+3 LCD: x+3 (3x-4) + (2x-5) x+3

18. 3)Divide numerator by common denominator *Simplify by dividing common factors (3x-4) + (2x+5) x+3 5x+1 x+3

19. Steps for Graphing

20. 1) Use shifts and transformations to graph horizontal asymptotes. f(x)= 2 - 3 X+1 Left 1 Down 3

21. 2) Make a table of values choose x values to the left and right of the vertical asymptote *x=0 needs to be a table value f(x)= 2 - 3 X+1 Left 1 Down 3 xY y -3--2.6 -2 -3 0 1 -3 2

22. 3) Plot points On white board...

23. 4) Identify: ●Domain ●Discontinuities ●Holes ●Vertical Asymptotes ●Horizontal Asymptotes ●x-intercept ●y-intercept f(x)= 2 - 3 X+1 Domain: All real numbers, x=-1 Discontinuities: x=-1 Holes: None VA: x=-1 HA: y=-3 X-Int: -⅓ Y-Int: (0,-1)

24. Practice Problems

25. r+5 - 6r -12 = 1 2r 2 -5r 2r 2 -5r 2r-5 4x 6 2x 4 5x 2 +15x = 10 X+3 x 2 -2x-3 x 2 +5x+4 4 5x + 5 12x-12 15x - 15 8r² x 6r³-9r² 6r³-9r² r-2