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Warm-Up 3/27. 1. G. Rigor: You will learn how to analyze, graph and solve equations of rational functions. Relevance: You will be able to use graphs and equations of rational functions to solve real world problems. . 2-5a Rational Functions. - PowerPoint PPT Presentation
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1.Warm-Up 3/27
G
𝑥=−𝑏2𝑎
Rigor:You will learn how to analyze, graph and solve
equations of rational functions.
Relevance:You will be able to use graphs and equations of rational functions to solve real world problems.
2-5a Rational Functions
A rational function is the quotient of two polynomial functions.
An asymptote is a line or curve that a graph approaches.
If a factor is removable, then there is a hole at that x value.
Example 1a: Find the domain of the function and the equations of the vertical or horizontal asymptotes, if any.𝑓 (𝑥 )= 𝑥+4
𝑥−3
Step 1 Find the domain. 𝑥−3≠0 𝑥≠3 (−∞ ,3)∪(3 ,∞ )
Step 2 Find the asymptotes, if any.(𝑥−3 )is not removable , so 𝑥=3a vertical asymptote .
Degree of numerator equals the degree of the denominator, so is the horizontal asymptote.
Check
Example 1b: Find the domain of the function and the equations of the vertical or horizontal asymptotes, if any.𝑔 (𝑥 )=8 𝑥2+5
4 𝑥2+1
Step 1 Find the domain. 4 𝑥2+1≠0 𝑥2≠− 14 (−∞ ,∞)
Step 2 Find the asymptotes, if any.Since domain is all realnumbers ,there are novertical asymptotes .
Degree of numerator equals the degree of the denominator, so therefore is the horizontal asymptote.
Check
Example 2a: Find the domain, the vertical or horizontal asymptotes and intercepts. Then graph the function.𝑔 (𝑥 )= 6
𝑥+3
Step 1 Find the domain. 𝑥+3≠0 𝑥≠−3 (−∞ ,−3)∪ (−3 ,∞)
Step 2 Find the asymptotes, if any.(𝑥+3 ) is not removable , so 𝑥=−3 avertical asymptote .
Degree of numerator less than the degree of the denominator, so is the horizontal asymptote.Step 3 There are no x-intercepts and (0, 2) is the y-intercept.
Step 4
Example 2b: Find the domain, the vertical or horizontal asymptotes and intercepts. Then graph the function.𝑘 (𝑥 )= 𝑥2−7 𝑥+10
𝑥−3
Step 1 Find the domain. 𝑥−3≠0 𝑥≠3 (−∞ ,3)∪ (3 ,∞)
Step 2 Find the asymptotes, if any.(𝑥−3 ) is not removable , so 𝑥=3a vertical asymptote .
Degree of numerator greater than the degree of the denominator, so no horizontal asymptote.Step 3 x-intercepts (5, 0) & (2, 0) and (0,) is the y-intercept.
Step 4 x y– 5 – 8.75
– 1 – 4.5
1 – 2
4 – 2
7 2.5
𝑘 (𝑥 )=(𝑥−5)(𝑥−2)(𝑥−3)
Example 3: Find the domain, the vertical or horizontal asymptotes and intercepts. Then graph the function.𝑓 (𝑥 )=3 𝑥2−3
𝑥2−9
Step 1 Find the domain. 𝑥2−9≠0 𝑥≠±3 {𝑥|𝑥≠±3 ,𝑥∈ℝ }
Step 2 Find the asymptotes, if any.(𝑥−3 ) , (𝑥+3 )are not removable , so 𝑥=±3 vertical asymptotes .
Degree of numerator is equal to the degree of the denominator, so 𝑦 = 3 is the horizontal asymptote.
Step 3 x-intercepts (– 1, 0) & (1, 0) and (0,) is the y-intercept.
Step 4 x y– 7 3.6
– 5 4.5
– 2 – 1.8
2 – 1.8
5 4.5
7 3.6
𝑓 (𝑥 )=3 (𝑥−1)(𝑥+1)(𝑥−3)(𝑥+3)
An oblique asymptote is a slant line that occurs when the degree of the numerator is exactly one more than the degree of the denominator.
Example 4: Find the domain, any asymptotes and intercepts. Then graph the function.𝑓 (𝑥 )= 2 𝑥3
𝑥2+𝑥−12
Step 1 Find the domain. 𝑥≠−4 𝑥≠3 {𝑥|𝑥≠−4 ,3 ,𝑥∈ℝ }Step 2 Find the asymptotes, if any.
(𝑥−3 ) , (𝑥+4 )are not removable , so 𝑥=−4 ,3 vertical asymptotes .Degree of numerator is exactly one more than the degree of the denominator, so No H. A. & 𝑦 = 2x – 2 is the oblique asymptote.
Step 3 x- & y-intercept (0, 0).
Step 4x y
– 7 – 22.87
– 6 – 24
– 5 – 31.25
– 3 9
1 – 0.2
4 16
5 13.889
6 14.4
𝑓 (𝑥 )= 2𝑥3
(𝑥−3)(𝑥+4)
Example 5: Find the domain, any asymptotes, holes and intercepts. Then graph the function.h (𝑥 )= 𝑥2−4
𝑥2−2 𝑥−8
Step 1 Find the domain. 𝑥≠4 𝑥≠−2 {𝑥|𝑥≠−2 ,4 ,𝑥∈ℝ }Step 2 Find the asymptotes, if any.
(𝑥−4 )is not removable , so 𝑥=4 vertical asymptotes .Degree of numerator is equal to the degree of the denominator, so 𝑦 = 1 is the horizontal asymptote.
Step 3 x-intercept (2, 0) and (0,) is the y-intercept.
Step 4x y
– 2 hole
– 1 .6
0 .5
1 .3333
2 0
3 – 1
4 V.A.
5 3
6 2
h (𝑥 )=(𝑥−2)(𝑥+2)(𝑥−4 )(𝑥+2)
√−1math!
2-5a Assignment: TX p138, 4-28 EOE
1.Warm-Up 3/26
A
√−1math!
2-5a Assignment: TX p138, 4-28 EOE