Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.
-
Upload
adele-blake -
Category
Documents
-
view
230 -
download
2
Transcript of Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.
![Page 1: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/1.jpg)
Unit 7—Rational Functions
Rational Expressions• Quotient of 2 polynomials
0,
91
132
3
52
15
3 2
3
2
pso
xory
xxor
y
xor
xy
yx
![Page 2: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/2.jpg)
Things to Consider
![Page 3: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/3.jpg)
Graphing Rational Functions
1. Factor2. Determine where discontinuities would occur in
the graph3. Graph any asymptotes on the graph and pick
points on both sides to locate the branches of the function
• If factors cancel, then you have a point of discontinuity (hole)
• If factors remain in the denominator, you have a vertical asymptote(s)
![Page 4: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/4.jpg)
Horizontal Asymptotes
1. If the degree of the numerator is bigger than the degree of the denominator, then there is NO H.A. (top-heavy)
2. If the degree of numerator is smaller than the degree of the denominator, then the HA is at y=0 (bottom heavy)
3. If the degree of numerator is equal to degree of the denominator, then the HA is equal to the leading coefficients (equal weight)
![Page 5: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/5.jpg)
Graphing Rational Functions
209
1272
2
xx
xxy
![Page 6: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/6.jpg)
Graphing Rational Functions
5
10133 2
x
xxy
![Page 7: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/7.jpg)
Graphing Rational Functions
2
322
2
xx
xxy
![Page 8: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/8.jpg)
Graphing Rational Functions
322
xx
xy
![Page 9: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/9.jpg)
Graphing Rational Functions
3
92
x
xy
![Page 10: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/10.jpg)
Graphing Rational Functions
1
92
2
x
xy
![Page 11: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/11.jpg)
To simplify a rational expression
• Look for common factors
3
3.
9124
94.
)1)(2(
)2(.
9
27.
2
2
4
3
x
xex
xx
xex
xx
xex
yx
yxex
![Page 12: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/12.jpg)
To simplify a rational expression
• Look for common factors
654
36.
25
5.
2
2.
2
2
xx
xex
x
xex
x
xex
![Page 13: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/13.jpg)
Multiply Rational Expressions
• Factor, Reduce common factors first, then multiply
3
4
127
)3(.
3
44
4
9.
2
22
2
2
x
x
xx
xex
x
xx
x
xex
![Page 14: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/14.jpg)
Multiply Rational Expressions
• Factor, Reduce common factors first, then multiply
5
25
9
27.
8
16
4
4.
2
2
32
2
3
x
x
x
xex
a
b
bb
abex
![Page 15: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/15.jpg)
Dividing Rational Expressions
• Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply
1
3
12.
123
65
2412
63.
22
2
2
2
x
x
xx
xex
x
xx
x
xex
![Page 16: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/16.jpg)
Dividing Rational Expressions
• Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply
2
2
4
322
9
16
3
4.
84
9
2
93.
y
x
y
xex
x
x
x
xxex
![Page 17: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/17.jpg)
Dividing Rational Expressions
77
5
4
22
56
25.
2
2
2
2
3
x
xx
x
x
xx
xxex
![Page 18: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/18.jpg)
Dividing Rational Expressions
bybxayax
bybxayax
bybxayax
bybxayaxex
.
![Page 19: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/19.jpg)
Dividing Rational Expressions
1077856
149
2
2
2
2
xxxxxxxx
![Page 20: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/20.jpg)
Dividing Rational Expressions
12 11
1
1 x
x
x
x
x
x
![Page 21: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/21.jpg)
Adding and Subtracting Rational Expressions
• To add fractions, you must have a common denominator
• To determine the LCD, list any common factor that occurs in two or more of the denominators only once in the LCD and then include all other factors that are not common.
![Page 22: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/22.jpg)
Adding and Subtracting Rational Expressions
1
4
1
52.
44
1.
22
x
x
x
xex
x
x
x
xex
![Page 23: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/23.jpg)
Adding and Subtracting Rational Expressions
9
4
3
35.
52
4
13
6.
2
x
x
x
xex
x
x
x
xex
![Page 24: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/24.jpg)
Adding and Subtracting Rational Expressions
)1)(1(
442.
168
3
16
4.
222
xxx
x
xx
xex
xxxex
![Page 25: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/25.jpg)
Adding and Subtracting Rational Expressions
2
3
2
87.
1
3
1
5
2
1.
22
yyyy
yex
xx
x
xex
![Page 26: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/26.jpg)
Adding and Subtracting Rational Expressions
324
7
245
23.
25
5
25
102.
2222
yyyy
yex
x
x
x
xex
![Page 27: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/27.jpg)
Complex Rational Expressions
x
yx1
1
11
![Page 28: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/28.jpg)
Complex Rational Expressions
x
xyx
11
24
2
![Page 29: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/29.jpg)
Complex Rational Expressions
263
1235
mm
mm
![Page 30: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/30.jpg)
Complex Rational Expressions
yx
yx11
11
![Page 31: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/31.jpg)
Complex Rational Expressions
6
3x
x
xx
![Page 32: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/32.jpg)
Solving Rational Equations
1. Factor all denominators2. Multiply both sides of equation by LCD3. Solve4. Eliminate any solution that would make the
denominator zero5. Check remaining solutions
![Page 33: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/33.jpg)
Solving Rational Equations
xxxx 3
8
3
452
![Page 34: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/34.jpg)
Solving Rational Equations
x
x 42
5
1
![Page 35: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/35.jpg)
Solving Rational Equations
45103
22
x
x
xx
x
![Page 36: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/36.jpg)
Solving Rational Equations
3
2
3
1
yy
y
![Page 37: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/37.jpg)
Solving Rational Equations
3
2
1
3
mm
![Page 38: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/38.jpg)
Solving Rational Equations
xxx
4
2
5050
![Page 39: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/39.jpg)
Solving Rational Equations
2
5
4
2
2
32
yy
y
y
![Page 40: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/40.jpg)
Solving Rational Inequalities
1. State the excluded values2. Solve the related equation3. Use those values on a number line and test
the values
![Page 41: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/41.jpg)
Solving Rational Inequalities
3)3(2
2
x
x
x
x
![Page 42: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/42.jpg)
Solving Rational Inequalities
12
4
c
![Page 43: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/43.jpg)
Graphing Rational Functions
Possible Graphs:
![Page 44: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/44.jpg)
Direct and Inverse Variation
• Direct Variation can be expressed in the form y=kx
• K is the constant of variation
• Equation of variation—equation representing the relationship between the variable but substitute the value of k
Ex. Y=2x (if k=2)
![Page 45: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/45.jpg)
Direct and Inverse Variation
10y when x Find
3. x12,y When ith x.directly w varies
y
![Page 46: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/46.jpg)
Direct and Inverse Variation
1y x when Find
5. x25,y When ith x.directly w varies
y
![Page 47: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/47.jpg)
Direct and Inverse Variation
4p when q Find
.3p 8,q When p. of squareith directly w varies
Q
![Page 48: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/48.jpg)
Direct and Inverse Variation
3m when L Find
.2m ,2
1L When m. of cubeith directly w varies
L
![Page 49: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/49.jpg)
Inverse Variation
• Expressed as y=k/x
![Page 50: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/50.jpg)
Direct and Inverse Variation
4
3y when x Find
.5
2 x3,y When with x.inversely varies
y
![Page 51: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/51.jpg)
Direct and Inverse Variation
4y when x Find
.2
1 x2,y When x.of square with theinversely varies
y
![Page 52: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/52.jpg)
Direct and Inverse Variation
8y x when Find
.3- x6,y When with x.inversely varies
y
![Page 53: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/53.jpg)
Joint Variation
• When one quantity varies directly with the product of two or more other quantities
• Combination Variation—when one quantity varies directly with another quantity and inversely with the other quantity
![Page 54: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/54.jpg)
Joint or Combination Variation
68y when x Find
.32 x4,y When z. of square theandith x directly w varies
zand
zandy
![Page 55: Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.](https://reader036.fdocuments.us/reader036/viewer/2022081508/56649edb5503460f94beb76c/html5/thumbnails/55.jpg)
Joint or Combination Variation
65y when x Find
.14 x20,y When z. of square the
withinversely andith x directly w varies
zand
zand
y