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 2 +8 has a degree of three, since the highest exponent = 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 2 +8 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…

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 2 +8 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

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

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 )

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…

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

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 …

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

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.

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 = 0−1 0 2 −0−6 = 1 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 ) Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes. 𝑓 0 = 0−1 0 2 −0−6 = 1 6 𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4

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 = 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 = 0−1 0 2 −0−6 = 1 6 𝑓 −4 = −4−1 (−4) 2 −(−4)−6 =−0.4 𝑓 −3 = −3−1 (−3) 2 −(−3)−6 =−0.7

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 ) Now test values on each interval. I like to find the y – intercept and stay close to the vertical asymptotes. 𝑓 0 = 0−1 0 2 −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

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 = 0−1 0 2 −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

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 = 0−1 0 2 −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

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 = 0−1 0 2 −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

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 = 0−1 0 2 −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

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…

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

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 )

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…

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

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 …

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.

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

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

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

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

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

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

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

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

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…

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

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 )

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…

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

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.

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…

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…

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

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

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+1 + 2 𝑥−1 where 2 𝑥−1 is a remainder 1 1 + 1 1 2

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+1 + 2 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 …

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+1 + 2 𝑥−1 where 2 𝑥−1 is a remainder 1 1 + 1 1 2 The oblique asymptote will be 𝑦=(𝑥+1)

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+1 + 2 𝑥−1 where 2 𝑥−1 is a remainder 1 1 + 1 1 2 Graph it… The oblique asymptote will be 𝑦=(𝑥+1)

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 = 0 2 +1 0−1 =−1

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 = 0 2 +1 0−1 =−1 𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7

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 # 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 = 0 2 +1 0−1 =−1 𝑓 −2 = (−2) 2 +1 (−2)−1 =−1.7 𝑓 0.5 = (0.5) 2 +1 (0.5)−1 =−2.5

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 # 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 = 0 2 +1 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

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 = 0 2 +1 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

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 = 0 2 +1 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

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…

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…