Upload
adele-blake
View
233
Download
2
Tags:
Embed Size (px)
Citation preview
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
Things to Consider
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)
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)
Graphing Rational Functions
209
1272
2
xx
xxy
Graphing Rational Functions
5
10133 2
x
xxy
Graphing Rational Functions
2
322
2
xx
xxy
Graphing Rational Functions
322
xx
xy
Graphing Rational Functions
3
92
x
xy
Graphing Rational Functions
1
92
2
x
xy
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
To simplify a rational expression
• Look for common factors
654
36.
25
5.
2
2.
2
2
xx
xex
x
xex
x
xex
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
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
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
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
Dividing Rational Expressions
77
5
4
22
56
25.
2
2
2
2
3
x
xx
x
x
xx
xxex
Dividing Rational Expressions
bybxayax
bybxayax
bybxayax
bybxayaxex
.
Dividing Rational Expressions
1077856
149
2
2
2
2
xxxxxxxx
Dividing Rational Expressions
12 11
1
1 x
x
x
x
x
x
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.
Adding and Subtracting Rational Expressions
1
4
1
52.
44
1.
22
x
x
x
xex
x
x
x
xex
Adding and Subtracting Rational Expressions
9
4
3
35.
52
4
13
6.
2
x
x
x
xex
x
x
x
xex
Adding and Subtracting Rational Expressions
)1)(1(
442.
168
3
16
4.
222
xxx
x
xx
xex
xxxex
Adding and Subtracting Rational Expressions
2
3
2
87.
1
3
1
5
2
1.
22
yyyy
yex
xx
x
xex
Adding and Subtracting Rational Expressions
324
7
245
23.
25
5
25
102.
2222
yyyy
yex
x
x
x
xex
Complex Rational Expressions
x
yx1
1
11
Complex Rational Expressions
x
xyx
11
24
2
Complex Rational Expressions
263
1235
mm
mm
Complex Rational Expressions
yx
yx11
11
Complex Rational Expressions
6
3x
x
xx
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
Solving Rational Equations
xxxx 3
8
3
452
Solving Rational Equations
x
x 42
5
1
Solving Rational Equations
45103
22
x
x
xx
x
Solving Rational Equations
3
2
3
1
yy
y
Solving Rational Equations
3
2
1
3
mm
Solving Rational Equations
xxx
4
2
5050
Solving Rational Equations
2
5
4
2
2
32
yy
y
y
Solving Rational Inequalities
1. State the excluded values2. Solve the related equation3. Use those values on a number line and test
the values
Solving Rational Inequalities
3)3(2
2
x
x
x
x
Solving Rational Inequalities
12
4
c
Graphing Rational Functions
Possible Graphs:
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)
Direct and Inverse Variation
10y when x Find
3. x12,y When ith x.directly w varies
y
Direct and Inverse Variation
1y x when Find
5. x25,y When ith x.directly w varies
y
Direct and Inverse Variation
4p when q Find
.3p 8,q When p. of squareith directly w varies
Q
Direct and Inverse Variation
3m when L Find
.2m ,2
1L When m. of cubeith directly w varies
L
Inverse Variation
• Expressed as y=k/x
Direct and Inverse Variation
4
3y when x Find
.5
2 x3,y When with x.inversely varies
y
Direct and Inverse Variation
4y when x Find
.2
1 x2,y When x.of square with theinversely varies
y
Direct and Inverse Variation
8y x when Find
.3- x6,y When with x.inversely varies
y
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
Joint or Combination Variation
68y when x Find
.32 x4,y When z. of square theandith x directly w varies
zand
zandy
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