1. OTHER RATIONAL FUNCTIONS
The “degree” of a function is the highest exponent that appears in the function. For example, 𝑓 𝑥 = 𝑥 3 −2 𝑥 has a degree of three, since the highest exponent = 3.

2. OTHER RATIONAL FUNCTIONS
The “degree” of a function is the highest exponent that appears in the function. For example, 𝑓 𝑥 = 𝑥 3 −2 𝑥 has a degree of three, since the highest exponent = 3.
More complicated rational functions have one or both of the polynomials with a degree higher than one. We will use the following chart to graph these functions…

3. OTHER RATIONAL FUNCTIONS
The “degree” of a function is the highest exponent that appears in the function. For example, 𝑓 𝑥 = 𝑥 3 −2 𝑥 has a degree of three, since the highest exponent = 3.
More complicated rational functions have one or both of the polynomials with a degree higher than one. We will use the following chart to graph these functions…
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator

4. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6

5. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )

6. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Graph them…

7. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Degree of numerator = 1
Degree of denominator = 2
n < k so x – axis is horizontal asymptote

8. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Degree of numerator = 1
Degree of denominator = 2
n < k so x – axis is horizontal asymptote
Graph it …

9. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
** these are the graphs where the sketch could cross the horizontal asymptote

10. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.

11. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6

12. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4

13. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7

14. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7
𝑓 −1 = −1−1 (−1) 2 −(−1)−6 =0.5

15. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7
𝑓 −1 = −1−1 (−1) 2 −(−1)−6 =0.5
𝑓 1 = 1−1 (1) 2 −(1)−6 =0

16. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 2 = 2−1 (2) 2 −(2)−6 =−0.25
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7
𝑓 −1 = −1−1 (−1) 2 −(−1)−6 =0.5
𝑓 1 = 1−1 (1) 2 −(1)−6 =0

17. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 2 = 2−1 (2) 2 −(2)−6 =−0.25
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 4 = 4−1 (4) 2 −(4)−6 =0.5
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7
𝑓 −1 = −1−1 (−1) 2 −(−1)−6 =0.5
𝑓 1 = 1−1 (1) 2 −(1)−6 =0

18. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 0− −0−6 = 1 6
𝑓 2 = 2−1 (2) 2 −(2)−6 =−0.25
𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4
𝑓 4 = 4−1 (4) 2 −(4)−6 =0.5
𝑓 5 = 5−1 (5) 2 −(5)−6 =0.3
𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7
𝑓 −1 = −1−1 (−1) 2 −(−1)−6 =0.5
𝑓 1 = 1−1 (1) 2 −(1)−6 =0

19. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 1 : Graph 𝑓 𝑥 = 𝑥−1 𝑥 2 −𝑥−6 = 𝑥−1 𝑥−3 𝑥+2
Roots of the denominator are : 𝑥=3,−2 ( vertical asymptotes )
Now sketch the graph for each interval…

20. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2

21. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )

22. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Graph them…

23. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 2
n = k so 𝑦= 𝑎 𝑐 = 2 1 =2 is horizontal asymptote

24. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 2
n = k so 𝑦= 𝑎 𝑐 = 2 1 =2 is horizontal asymptote
Graph it …

25. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.

26. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0

27. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1

28. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6

29. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6
𝑓 0.5 = 2 (0.5) 2 (0.5) 2 +(0.5)−2 =−0.4

30. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6
𝑓 0.5 = 2 (0.5) 2 (0.5) 2 +(0.5)−2 =−0.4
𝑓 −3 = 2 (−3) 2 (−3) 2 +(−3)−2 =4.5

31. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −4 = 2 (−4) 2 (−4) 2 +(−4)−2 =4.5
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6
𝑓 0.5 = 2 (0.5) 2 (0.5) 2 +(0.5)−2 =−0.4
𝑓 −3 = 2 (−3) 2 (−3) 2 +(−3)−2 =4.5

32. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −4 = 2 (−4) 2 (−4) 2 +(−4)−2 =4.5
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 2 = 2 (2) 2 (2) 2 +(2)−2 =2
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6
𝑓 0.5 = 2 (0.5) 2 (0.5) 2 +(0.5)−2 =−0.4
𝑓 −3 = 2 (−3) 2 (−3) 2 +(−3)−2 =4.5

33. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes.
𝑓 0 = 2 (0) 2 (0) 2 +(0)−2 =0
𝑓 −4 = 2 (−4) 2 (−4) 2 +(−4)−2 =4.5
𝑓 −1 = 2 (−1) 2 (−1) 2 +(−1)−2 =−1
𝑓 2 = 2 (2) 2 (2) 2 +(2)−2 =2
𝑓 −1.5 = 2 (−1.5) 2 (−1.5) 2 +(−1.5)−2 =−3.6
𝑓 3 = 2 (3) 2 (3) 2 +(3)−2 =1.8
𝑓 0.5 = 2 (0.5) 2 (0.5) 2 +(0.5)−2 =−0.4
𝑓 −3 = 2 (−3) 2 (−3) 2 +(−3)−2 =4.5

34. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 2 : Graph 𝑓 𝑥 = 2 𝑥 2 𝑥 2 +𝑥−2 = 2 𝑥 2 𝑥+2 𝑥−1
Roots of the denominator are : 𝑥=−2, +1 ( vertical asymptotes )
Now sketch the graph for each interval…

35. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1

36. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )

37. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Graph it…

38. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote

39. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.

40. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…

41. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division…

42. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1 + 1

43. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1 1 1 + 1 1 2

44. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1
This gives us :
𝑥 𝑥−1
where 2 𝑥−1 is a remainder 1 1 + 1 1 2

45. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1
This gives us :
𝑥 𝑥−1
where 2 𝑥−1 is a remainder 1 1 + 1 1 2
We will use only the (𝑥+1) part of the answer …

46. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1
This gives us :
𝑥 𝑥−1
where 2 𝑥−1 is a remainder 1 1 + 1 1 2
The oblique asymptote will be 𝑦=(𝑥+1)

47. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Degree of numerator = 2
Degree of denominator = 1
n > k so no horizontal asymptote
There is however an oblique ( slanted ) asymptote.
These occur when n > k…
To find the oblique asymptote, use synthetic division… 1 1 1
This gives us :
𝑥 𝑥−1
where 2 𝑥−1 is a remainder 1 1 + 1 1 2
Graph it…
The oblique asymptote will be 𝑦=(𝑥+1)

48. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1

49. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1
𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7

50. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1
𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7
𝑓 0.5 = (0.5) 2 +1 (0.5)−1 =−2.5

51. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1
𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7
𝑓 0.5 = (0.5) 2 +1 (0.5)−1 =−2.5
𝑓 1.5 = (1.5) 2 +1 (1.5)−1 =6.5

52. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1
𝑓 2 = (2) 2 +1 (2)−1 =5
𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7
𝑓 0.5 = (0.5) 2 +1 (0.5)−1 =−2.5
𝑓 1.5 = (1.5) 2 +1 (1.5)−1 =6.5

53. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Once again, I like to find the y – intercept and choose some values near any vertical asymptote…
𝑓 0 = −1 =−1
𝑓 2 = (2) 2 +1 (2)−1 =5
𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7
𝑓 4 = (4) 2 +1 (4)−1 =5.7
𝑓 0.5 = (0.5) 2 +1 (0.5)−1 =−2.5
𝑓 1.5 = (1.5) 2 +1 (1.5)−1 =6.5

54. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Now sketch your graph on each interval…

55. OTHER RATIONAL FUNCTIONS
Let 𝑓 𝑥 = 𝑎 𝑥 𝑛 + ∙∙∙ 𝑐 𝑥 𝑘 + ∙∙∙ be a rational function whose numerator has degree n and whose denominator has degree k .
If 𝑛=𝑘 , then the line 𝑦=𝑎/𝑐 is a horizontal asymptote
If 𝑛<𝑘 , then the x – axis is a horizontal asymptote
If 𝑛>𝑘 , then there is no horizontal asymptote ( it will be oblique )
** vertical asymptotes will still be roots of the denominator
EXAMPLE # 3 : Graph 𝑓 𝑥 = 𝑥 2 +1 𝑥−1
Roots of the denominator are : 𝑥=1 ( vertical asymptotes )
Now sketch your graph on each interval…