Unit!7:!Radicals!and!Rationals! · 2018. 9. 28. · Unit!7:!Radicals!and!Rationals!!!...

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


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


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. !!!!!!!!!!"!!!"

.


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


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


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


Unit!7:!Radicals!and!Rationals! · 2018. 9. 28. · Unit!7:!Radicals!and!Rationals!!!...

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Transcript of Unit!7:!Radicals!and!Rationals! · 2018. 9. 28. · Unit!7:!Radicals!and!Rationals!!!...