1. Removable Discontinuities & Vertical Asymptotes
Rational Functions:
Removable Discontinuities & Vertical Asymptotes

2. What is a Rational function?
A rational function is a ratio of two polynomial functions where the polynomial in the denominator does not equal zero

3. Rational Functions First, always factor if possible.
Then, divide out common terms.

4. Are they the same??
No!

5. Removable Discontinuity (hole)
a single point in which the function has no value
---A hole produces a case where the rational function has for particular x values.
---we want to look for cases where you can cancel a factor from the numerator and the denominator of the rational function

6. Write the equation:

7. Write the Equation:
Write an equation for a function that is a horizontal line with a constant height of 3 but with a hole at x = -2.
The same as #1 but with holes at x = -2 and x = 5.
Write an equation for a graph that looks like the parabola , but has holes at x = 1 and x = 3.

8. Vertical Asymptotes Where are the vertical asymptotes?
How do we know there is a vertical asymptote by looking at the equation?

9. Vertical Asymptotes:
A non-removable discontinuity that produces a vertical line in which the function approaches but never touches
A VA produces a case where the rational function has 𝑐 0 for particular x values. (where c is a constant)
We want to look for cases where the denominator is equal to zero (after canceling factors that contribute to holes)

10. Write the Equation:
4) Come up with the equation for a function with two vertical asymptotes, at x = -2 and x = 5. 5) The same as #4 but now with vertical asymptotes at x = -2 and x = 5, and a hole at x =3.

11. Domain
What is going to affect the domain of our functions? -Removable discontinuities -Vertical asymptotes

12. Put it all together!
Find all holes, vertical asymptotes, and the domain for each problem.
𝑓 𝑥 = 2 𝑥 2 +13𝑥+15 𝑥 2 +13𝑥+40
𝑓 𝑥 = 2 𝑥 2 +11𝑥+5 5 𝑥 2 −11𝑥+2 6) 8)
𝑓 𝑥 = 2 𝑥 2 −4𝑥 𝑥 2 +3𝑥
𝑓 𝑥 = 𝑥 2 +4𝑥+3 𝑥 3 − 𝑥 2 −2𝑥 7) 9)

13. Homework:
WB p #1-7 Directions: Find the holes, vertical asymptotes, and state the domain.