Monday March 21, 2016
Warm-up:
Simplify
1.
Factoring then simplify
2.
Monday March 21, 2016
Warm-up:
Simplify
1.
Factoring then simplify
2.
8.1 Multiply and Dividing Rational Expressions
Objective: To simply rational expressions
EQ: Can you reduce expressions in the form of a fraction?
What is rational function?
Simplify Rational Expression:
Example 1:
Solution:
a.
b.
Example 2:
Solution:
Since 4x(x2 +6x +8) = 4x(x+2) (x+4), therefore the expression is
undefined when x = 0, -2, and -4
Practice:
8.1 Multiply and Dividing Rational Expressions
Objective: To simply rational expressions
EQ: Can you reduce expressions in the form of a fraction?
What is rational function?
Simplify Rational Expression:
Example 1:
Solution:
a.
b.
Example 2:
Solution:
Since 4x(x2 +6x +8) = 4x(x+2) (x+4), therefore the expression is
undefined when x = 0, -2, and -4
Practice:
8.1 Multiply and Dividing Rational Expressions
Example 3:
Solution:
Example 4: Simplify
Solution:
Example 3:
Solution:
8.2 Adding & Subtracting Rational Expressions
Obj: To determine the LCM then add or subtract rational expressions
EQ: How do you find the LCM and LCD
To find the LCM of 2 or more polynomials: First factor them ,
then take the greatest number of times of each factor .
Example: Find the LCM of each set of polynomials
To add or subtract rational expressions:
1. Find LCD
2. Rewrite each rational expression in term of LCD.
Example:
8.3 Graphing Reciprocal Functions
Obj: To identify the asymptotes, domain, and range of a function
EQ: Can you identify a, h, and k ?
What is reciprocal function?
Parent Function of Reciprocal Functions:
Transformations of Reciprocal Functions
8.3 Graphing Reciprocal Functions
Example 1: Identify the asymptotes, domain, and range:
Solution:
1. Asymptotes:
vertical x=3
horizontal y =0
2.
Domain: D = { x | x ≠ 3 }
or (-∞, 3) U (3, +∞)
3.
Range: R = { x| x ≠ 0}
or (-∞, 0) U (0, +∞)
Example 2: Identify the asymptotes, domain, and range:
8.3 Graphing Reciprocal Functions
Example 3: Identify the asymptotes, domain, and range:
Example 4: Identify the asymptotes, domain, and range:
8.4 Graphing Rational Functions
Obj: To identify the asymptotes, domain, and range of a function
EQ: Can you identify a, h, and k ?
Vertical and Horizontal Asymptotes
Examples: Identify the asymptotes:
No horizontal asymptote
VA:
x = -1
One horizontal asymptote
VA:
x = -1, x = 1
HA:
y=0
VA:
x=3
HA:
y=2
Example 3: Identify the asymptotes, domain, and range:
Example 4: Identify the asymptotes, domain, and range:
8.3 Graphing Reciprocal Functions
Oblique Asymptotes and Point Discontinuity:
Graph with Point Discontinuity:
Solution:
8.5 Variation Functions
Obj: To recognize and solve direct, inverse variation functions
EQ: What is a constant of variation k?
Different forms of variation functions:c
1.
Direct Variation: y = kx
(y increases when x increases)
2.
Joint Variation:
(y increases when x&z ncrease)
3.
Inverse Variation: y = k/x
y = kxz
Example 1: Direct Variation
Example 2: Inverse Variation
(y decreases when x increases)
8.5 Variation Functions
Example 3:
Example 4: Which table shows an inverse variation?
8.5 Variation Functions
Example 3:
Example 4: Inverse Variation
8.6 Solving Rational Equations
Obj: To solve rational equations
EQ: Can you still simplify rational expressions?
What is a rational equation?
An equation that contains one or more rational expressions
How to solve a rational equation?
It would be easier to solve a rational equation once the fractions
are eliminated.
Example 1:
Solution:
1. Determine the LCD for these 3 fractions: 18(x+3)
2. Multiply the equation with LCD:
3. Simplify:
4. Distribute, combine like terms then solve for x:
8.6 Solving Rational Equations
Obj: To solve rational equations
EQ: Can you still simplify rational expressions?
What is a rational equation?
An equation that contains one or more rational expressions
How to solve a rational equation?
It would be easier to solve a rational equation once the fractions
are eliminated.
Example 1:
Solution:
1. Determine the LCD for these 3 fractions: 18(x+3)
2. Multiply the equation with LCD:
3. Simplify:
4. Distribute, combine like terms then solve for x:
8.6 Solving Rational Equations
Example 2:
Solution:
1. Determine the LCD for these 3 fractions: (x+3) (x+5)
2. Multiply the equation with LCD:
3. Simplify:
4. Distribute, combine like terms then solve for x:
5. Check:
Practice:
8.6 Solving Rational Equations
Example 2:
Solution:
1. Determine the LCD for these 3 fractions: (x+3) (x+5)
2. Multiply the equation with LCD:
3. Simplify:
4. Distribute, combine like terms then solve for x:
5. Check:
Practice: