1. 2.5 – Rational Functions

2. Rational Functions
𝑓 𝑥 = 𝑎(𝑥) 𝑏(𝑥) *Check for Vertical and Horizontal Asymptotes!

4. Example 1:
Find the domain and the equations of the vertical or horizontal asymptotes, if any. a. b.

5. Graphs of Rational Functions
Vertical Asymptotes: May occur at the real zeros of the denominator.
Horizontal Asymptotes: Compare degree of num. (n) and degree of denom. (m):
n < m: HA is y = 0
n = m: HA is 𝑦= 𝑎 𝑛 𝑏 𝑚
n > m: no HA

6. Graphs of Rational Functions
x-intercepts: Occur at real zeros of numerator.
y-intercepts: Value of the function when x = 0.

7. Example 2:
For , determine any vertical and horizontal asymptotes and intercepts. Then graph the function and state its domain.

8. Example 3:
For , determine any vertical and horizontal asymptotes and intercepts. Then graph the function and state its domain.

9. Holes: Removable discontinuities.
When numerator & denominator have common factors. (Any factors that cancel.)
Holes are at the zeros of the common factors.
Find the ordered pair at that x-value.

10. Holes:

11. Example 4:
Determine the vertical and horizontal asymptotes and holes of the graph of
A. vertical asymptote at x = –2, horizontal asymptote at y = –2; no holes
B. vertical asymptotes at x = –5 and x = –2; horizontal asymptote at y = 1; hole at (–5, 3)
C. vertical asymptotes at x = –5 and x = –2; horizontal asymptote at y = 1; hole at (–5, 0)
D. vertical asymptote at x = –2; horizontal asymptote at y = 1; hole at (–5, 3)

12. Solving Rational Equations
Find LCD & multiply each term in the equation by the LCD.
Check for extraneous solutions.

13. Example 5:
Solve:

14. Example 6:
Solve:

15. Example 7:
Solve:

16. Example 8:
Suppose the current I, in amps, in an electric circuit is given by the formula , where t is time in seconds. At what time is the current 2 amps?

17. Exit Slip:
Solve: