Unit 7: Radicals and Rationals
Day 1: Square Root Functions
Day 2: Simplifying Radical Expressions
Day 3: Operations with Radical Expressions
Day 4: Radical Equations
Day 5:Rational Functions
Day 6: Simplifying Rational Expressions
Day 7: Simplifying Rational Expressions
Day 8: Multiplying and Dividing Rational Expressions
Day 9: Multiplying and Dividing Rational Expressions
Day 10: Adding and Subtracting Rational Expressions
Day 11: Adding and Subtracting Rational Expressions
Day 12: Rational Equations
Day 13: Rational Equations
Day 14: Review
Day 15: Review
Day 16: Test
Square Root Functions
A square root function contains the _____________ of a variable.
Square root functions are a type of ___________ function.
In order for a square root to be a real number, the ____________, or the expression under the
radical sign, cannot be ____________.
Values that make the radicant negative are not included in the _______________.
Parent Function:
Shape of Graph:
Domain:
Range:
Radical functions, like quadratic functions, can be _____________ horizontally and vertically,
____________ by a scale factor, and ___________ across the x or y-‐axis.
Horizontal Translation:
Vertical Translation:
Dilation:
Reflection:
Examples:
Graph 𝑦 = 3 𝑥 − 2 and compare it to the parent graph.
State the domain and range.
Graph 𝑦 = − 𝑥 + 1 and compare to the parent graph.
State the domain and range.33
Practice:
Graph each function, and compare to the parent graph. Be sure to label each function. Include 4 graphs on each coordinate plane. State the domain and range.
1. 𝑦 = 2 𝑥 − 3
Domain: Range:
2. 𝑦 = −𝑥 + 4
Domain: Range:
3. 𝑦 = 𝑥 − 3− 2
Domain: Range:
4. 𝑦 = − 𝑥 + 6
Domain: Range:
5. 𝑦 = !!𝑥 + 1
Domain: Range:
6. 𝑦 = − −𝑥
Domain: Range:
7. 𝑦 = 3 𝑥 + 5− 4
Domain: Range:
8. 𝑦 = − 𝑥 − 1+ 7
Domain: Range:
Homework: Glencoe Quiz 10.1
Simplifying Radical Expressions
Product Property of Square Roots:
Quotient Property of Square Roots:
Examples:
Simplify 180.
Simplify 120𝑎!𝑏!𝑐!.
Simplify !"!".
Simplify !"!!!!"!!
.
Practice:
Simplify each expression.
1. 75 6. !!"
2. 20𝑎!𝑏! 7. ! !! !
3. 45𝑥!𝑦!𝑧! 8. !""!!
!""!!
4. 4 10 · 3 6 9. !!
!!
5. 72𝑎!𝑏!𝑐! 10. !"!!
!!
Homework: Glencoe Quiz 10.2
Operations with Radical Expressions
Adding and Subtracting:
Multiplying:
Examples:
Simplify 10 6− 5 3+ 6 3− 4 6
Simplify 3 12+ 5 75
Multiply 3 2− 2 5 4 20+ 8
Practice:
Simplify or multiply out each expression
1. 2 5+ 6+ 4 5− 4 6
2. 20+ 2 5− 3 5
3. 2 3+ 4 5
4. 3 2 3 7+ 2 5
5. 8𝑎 − 2𝑎 + 5 2𝑎
6. 5− 18 7 5+ 4 3
7. 2+ 3 3 12+ 2 6
8. 80− 20+ 180
9. 50+ 2 18− 75+ 3 27
10. 3+ 2 6!
Homework: Glencoe 10.3 Quiz
Radical Equations
Equations containing radicals with variables in the radicand are called radical equations. These can be solved by first using the following steps.
Step 1:
Step 2:
Step 3:
Squaring each side of an equation sometimes produces ________________, or solutions that are not solutions of the original equation. Therefore, it is very important that you check each solution.
Examples:
4𝑥 − 7+ 2 = 7 16 = !! 𝑥 + 3 = 𝑥 − 3
Practice:
1. 4𝑥 − 1 = 3
2. 3𝑏 − 2+ 19 = 24
3. 3𝑟 + 2 = 2 3
4. 4𝑥 − 4 = 𝑥
5. 3𝑥! + 12𝑥 + 1
Homework: Glencoe 10.4 Quiz
Rational Functions
The function 𝑦 = !"! is an example of a __________. Because division by zero is undefined, any
value of a variable that results in a denominator of zero must be excluded from the domain of
that variable. These are called ________________ of the rational function.
Because excluded vales are undefined, they affect the graph of the function. An ____________
is a line that the graph of a function approaches but does not touch. A rational function in the
form 𝑦 = !!!!
+ 𝑐 has a vertical asymptote at the x-‐value that makes the denominator equal
zero, ________. It has a horizontal asymptote at y = c.
Examples:
Identify the excluded value in the equation 𝑦 = !!!!
.
Identify the asymptotes then graph 𝑦 = !!!!
+ 2.
Practice:
Identify the excluded values for each equation.
1. 𝑦 = !!!!
2. 𝑦 = !!!!!!
3. 𝑦 = !!!!!"
Identify the asymptotes and graph each equation.
4. 𝑦 = !!+ 1 5. 𝑦 = !
!!! 6. 𝑦 = !
!!!− 2
Homework: Glencoe 11.2 Quiz
Simplifying Rational Expressions
Rational Expression:
To simplify a rational expression, first ____________ the numerator and denominator. Then
divide each by the ___________________________.
Examples:
Simplify !"!!
!"!".
Simplify !!!!!!!!!!!
.
Practice:
Simplify each expression. State the excluded values of the variables.
1. !"!"!!!!
.
2. !!!!!!!!!
.
3. !!!!!!!"!!!!!!!
.
4. !!!!!!!!!!
.
5. !!!!!!!!!!"!!!"
.
Homework: Glencoe 11.3 Quiz
Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions:
Dividing Rational Expressions:
Examples:
!!!!!!!!
· !!!!!"
!!!!"!!!!
· !!!!!!!!!!"
!"!!!!!!!!
÷ !!!!!"
!!!!!!!"!!!!!!!!"
÷ !!!!!!!!!
Practice:
1. 6𝑎𝑏𝑎!𝑏! ·
𝑎!
𝑏!
4. 3𝑥𝑦!
8 ÷6𝑥𝑦1
2. 𝑥 + 2𝑥 − 4 ·
𝑥 − 4𝑥 − 1
5. 2𝑛 − 42𝑛 ÷
𝑛! − 4𝑛
3. 8𝑥 + 8
𝑥! − 2𝑥 + 1 ·𝑥 − 12𝑥 + 2
6. 𝑎! + 7𝑎 + 12𝑎! + 3𝑎 − 10÷
𝑎! − 9𝑎! − 25
Homework: Glencoe 11.4 Quiz
Adding and Subtracting Rational Expressions
To add or subtract rational expressions you must have a _________________________. Then
simply add the _________________ and then write the sum over the denominator. If possible,
simplify the resulting rational expression.
Examples:
𝑛 + 3𝑛 +
8𝑛 − 44𝑛
3𝑥𝑥! − 4𝑥 −
1𝑥 − 4
Practice:
1. 1𝑎 +
73𝑎
4. 59𝑥 −
1𝑥!
2. 84𝑎! +
63𝑎
5. 𝑦
𝑦 − 3−3
𝑦 + 3
3. 𝑦 + 2
𝑦! + 5𝑥 + 6+2− 𝑦
𝑦! + 𝑦 − 6
6. 𝑞
𝑞! − 16−𝑞 + 1
𝑞! + 5𝑞 + 4
Homework: Glencoe 11.6 Quiz
Rational Functions and Equations
To solve equations containing rational expressions first eliminate the denominators, then continue to solve as normal.
Examples:
𝑥 − 33 +
𝑥2 = 4
15𝑥! − 1 =
52(𝑥 − 1)
Practice:
Solve each equation. State any extraneous solutions.
1. 3𝑥 =
6𝑥 + 1
4. 𝑚 + 4𝑚 +
𝑚3 =
𝑚3
2. 𝑞 + 4𝑞 − 1+
𝑞𝑞 + 1 = 2
5. 𝑥 − 15 =
2𝑥 − 215
3. 4𝑧
𝑧! + 4𝑧 + 3 =6
𝑧 + 3+4
𝑧 + 1
6. 𝑥! − 16𝑥 − 4 + 𝑥! = 16
Homework: Glencoe 11.8 Quiz