1. Pre-Calculus Chapter 2 section 6 1 of 14 Warm - up

2. Rational Functions and Asymptotes

3. Pre-Calculus Chapter 2 section 6 3 of 14 Essential Question How do you find the domain and asymptote of a rational function? Key Vocabulary: Rational Function

4. Pre-Calculus Chapter 2 section 6 4 of 14 Rational Function A rational function is a quotient of two or more polynomial functions. It has the form of The parent graph is Notice the branches that approach the y-axis but never crosses. The line it approaches is called an asymptote. In this case we see x = 0 is a vertical asymptote.

5. Pre-Calculus Chapter 2 section 6 5 of 14 Asymptotes

6. Pre-Calculus Chapter 2 section 6 6 of 14 Let’s go Vertical Determine the asymptotes for the graph of Since f(1) is undefined there may be a vertical asymptote at x = 1. To verify we have to check from the left and right to see whether f(x) → ∞ or f(x) → – ∞ as x → 1. xf(x)f(x).9–9.99–99.999–999 1.111 1.01101 1.0011001 The values in the table suggest that f(x) → ∞ as x → 1, so x = 1 is a vertical asymptote. On the next slide we will find the horizontal asymptotes.

7. Pre-Calculus Chapter 2 section 6 7 of 14 Horizontal Asymptotes (HA) … the easy way If the degree of the denominator is bigger than the degree of the numerator, --- The HA is the x-axis (y = 0). If the degree of the numerator is bigger than the denominator, ---- --- There is no horizontal asymptote. If the degrees of the numerator and denominator are the same. --- The HA = the leading coefficient of the numerator divided by the leading coefficient of the denominator One way to remember this is the following pneumonic device: BOBO BOTN EATS DC BOBO - Bigger on bottom, y = 0 BOTN - Bigger on top, none EATS DC - Exponents are the same, divide coefficients

8. Pre-Calculus Chapter 2 section 6 8 of 14 Finding Asymptotes Find all the horizontal and vertical asymptotes for the graphs of: a.Horizontal: Degree is Bigger on the Bottom (BOBO) so y = 0 Vertical: 3x 2 + 1 = 0 has no Real solutions so No Vertical Asymptotes( VA) b.Horizontal: Degrees are equal – EATS DC so divide lead coefficients 2/1 = 2: y = 2 is HA Vertical:

9. Pre-Calculus Chapter 2 section 6 9 of 14 Shared Factors and holes There are times when the numerator and denominator of a rational function share a common factor. When x = 3 a denominator equals 0. You might expect a vertical asymptote at x = 3, however it is a common factor. Numerically f(x) has the same values as g(x) = x + 2 except at 3.

10. Pre-Calculus Chapter 2 section 6 10 of 14 Holes Because the values are the same. We get a hole in our function at the point (3, 5)…

11. Pre-Calculus Chapter 2 section 6 11 of 14 Find all the HA, VA, and holes in the graph of :. Horizontal: EAT DC so y = 1 Vertical: First we have to factor the top and bottom. So from this, x – 3 = 0 yield a VA of x = 3. Because x = –2 is not a VA there is a hole in the graph at x = –2. Next put –2 in for x in the simplified function to find the y coordinate of the hole.

12. Pre-Calculus Chapter 2 section 6 12 of 14 Find a. the domain of f, b. the HA and c. VA of the graph of: a. Because the denominator is zero when –3x 3 + 1 = 0, we solve this to find the domain is all real numbers except: b. HA: EATS DC c. VA we found when we set denominator = 0.

13. Pre-Calculus Chapter 2 section 6 13 of 14 Essential Question How do you find the domain and asymptote of a rational function?

14. Pre-Calculus Chapter 2 section 6 14 of 14 Daily Assignment CChapter 2 Section 6 TText Book PPg 152 – 153 ##1, 3, 7 – 12 all, 13 – 27 Odd RRead Section 2.7 SShow all work for credit.