Standards for Radical Functions

• MM1A2a. Simplify algebraic and numeric expressions involving square root.

• MM1A2b. Perform operations with square roots.

• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.

bxy

Radical Functions

• Essential questions:1. What is a radical function?2. What does the graph look like and how does

it move?3. How are they used in real life applications?

Real Life Applications

• Pythagorean Theorem• Distance Formula• Solving any equation that includes a

variable with an exponent, such as:

3

2

3

4rV

rA

Radical Expressions

• Index Radical Sign

Radicand

3 42 x

General Radical Equation

khxbay )(

Vertical stretch or compressionby a factor of |a|; for a < 0, the graph is a reflections across the x-axis

Vertical translation k unitsup for k > 0 and |k| unitsdown for k < 0

Horizontal stretch or compression by a factor of |1/b|; for b < 0, the graph is a reflection across the y-axis(b = 1 or -1 for this course)

Horizontal translation h units to the right for h > 0 and |h| units to the left if h < 0.(h = 0 for this course)

Radical Functions

• Make a (some) table(s), graph the followingfunctions and describe the transformations forx = 0, 1, 4, 9, 16 & 25.

• What transformationRules do you see fromYour graphs? )32()(

32)(

2)(

)(

xxi

xxh

xxg

xxf

What value for x gives us a zero under the radical?

That’s our smallest value in our t-chart.

X y

0 0

1 1

4 2

9 3

16 4

25 5

0

1

4

9

16

25

-1 1 3 5 7 9 11 13 15 17 19 21 23 25

-10

-8

-6

-4

-2

0

2

4

6

8

10

xxf )(

What value for x gives us a zero under the radical?

That’s our smallest value in our t-chart.

X y

0 0

1 2

4 4

9 6

16 8

25 10

02

12

42

92

162

252

xxf 2)(

-1 1 3 5 7 9 11 13 15 17 19 21 23 25

-10

-8

-6

-4

-2

0

2

4

6

8

10

What value for x gives us a zero under the radical?

That’s our smallest value in our t-chart.

X y

0 -3

1 -1

4 1

9 3

16 5

25 7

302

312

342

392

3162

3252

32)( xxf

-1 1 3 5 7 9 11 13 15 17 19 21 23 25

-10

-8

-6

-4

-2

0

2

4

6

8

10

What value for x gives us a zero under the radical?

That’s our smallest value in our t-chart.

X y

0 3

1 1

4 -1

9 -3

16 -5

25 -7

)32(1)( xxf

02

12

42

92

162

252

-1 1 3 5 7 9 11 13 15 17 19 21 23 25

-10

-8

-6

-4

-2

0

2

4

6

8

10

Radical Functions

• State an equation that would make the square root function shrink vertically by a factor of ½ and translate up 4 units.

• How would we reflect the above equation across the y-axis?

• Make the “x” negative

45.0 xy

Domain & Range: Radical Functions

• State the domain,range, and intervalsof increasing anddecreasing for each function.

)32()(

32)(

2)(

)(

xxi

xxh

xxg

xxf

Graphing Radical Functions Summary

• Transformations for radical functions are the same as polynomial functions.

• The domain of the parent function is limited to {x | x 0} (the set of all x such that x 0)

• The range of the parent function is limited to {y | y 0} (the set of all y such that y 0)

• The domain and range may change as a result of transformations.

• The parent radical function continuously increases from the origin.

Simplifying Radical Expressions• Square and square root

are inverse functions, butthe square root has to bepositive

oddisnifxx

evenisnifxx

therefore

xifxx

xifxx

n

n

____

____

0_,

0_,

2

2

525 5)5( 2 552

636 6)6( 2 662

Simplifying Radical Expressions

baab * abba *or

• “Simplify” a radical means to:1. Take all the perfect squares out of the

radicand.Simplify:

32 150 24*6 3*6

Simplifying Radical Expressions

• “Simplify” a radical means to:2. Combine terms with like radicands

• Must have the same radicand to be able to add or subtract radials

• Simplify:

8322 )3323(322

Simplifying Radical Expressions Quotient Property of Square Roots: If a ≥ 0 and

b > 0:

• “Simplify” a radical means to:3. Do not leave a radical in the denominator

• Simplify:

2

6

8

2

b

a

b

a

10

27

2

Simplify Expressions via Conjugates

•Remember: (a+ b) is the conjugate of (a – b)•We get conjugates when we factor a perfect square minus a perfect square.•We also get conjugates other times. Simplify:

)75)(75( )23)(23(

Simplify Expressions via Conjugates

Simplify:

85

3

55

53

Summary Important Operations • Square and square root

are inverse functions, butthe square root has to bepositive

• Product Property of Square Roots: If a ≥ 0 and b ≥ 0: • Quotient Property of Square Roots: If a ≥ 0 and

b > 0:

• b

a

b

a

baab *

0_,

0_,

2

2

xifxx

xifxx

Summary Simplification Rules• To “simplify” a radical means to:1. Take all the perfect squares out of the

radicand.2. Combine terms with like radicands3. Do not leave a radical in the denominator

Simplifying Radical Expressions

• Homework page 144, # 3 – 24 by 3’s and 25 & 26

Warm-upSimplify1.

2.

Solve.

3.

4.

4 7 125 80

3 3 2 3( )

x 2 0

4 7 5

3 6 3

4

3 2 3 7x 6

Standards for Radical Functions

• MM1A2a. Simplify algebraic and numeric expressions involving square root.

• MM1A2b. Perform operations with square roots.

• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.

bxy

Radical Functions

• Today’s essential questions:1. How do we find the solution of a radical

function?2. How are they used in real life applications?

12.3 Solving Radical Equations 1.Get the radical on one side.

2. Square both sides of the equals sign.

3. Solve for the variable.

4. Check your answer. IF the answer doesn’t check, then “no solution.”

EXAMPLE 3:

10 6 2 x100 6 2 x

10 6 17 2 ?

( ( ) )

102 6 x

x 17

10 10

EXAMPLE 1:

3 21 0x

3 21x 3 49 21 0

?

3 7 21 0( )?

42 0x 7x 49

21 21 0 ?

Your Turn – Solve with your

neighbor:

3 2 14x 3 12x

3 16 2 14 ?

3 4 2 14( )?

x 4x 16

12 2 14 ?

14 14

• The process is the same:1. Get a radical on one side.2. Square both sides of the equal sign.3. Solve for the variable.4. Repeat as necessary5. Check your answer. IF the answer does

not check, then there is NO SOLUTION for that answer

What if > One Radical?

Example 3:

19357 xx

19357 xx

244 x

6x

19)6(35)6(7

3737

1918542

?

?

Example 3:

242 xx

2422 xx

02422 xx

4

_6

x

orx

24)6(26

66

24126

?

?

046 xx24)4(24

?

?2484

164

Your Turn – Solve

Individually:

3212 xx

96212 2 xxx

01242 xx

2

_6

x

orx

3621)6(2

39 32112

?

?

026 xx ?

?

55

3221)2(2

5214

Practice with Tic-Tac-Toe• The object is to get three in a row.• Work together in designated pairs.• Notice: Different problems have

different points.• Your score will be the three in a row

you solve with the most points

Practice• Page 148, # 3 – 30 by 3’s and 31 & 32