Radical Functions and Equations L. Waihman 2002 radical radicand index.

Radical Functions and Equations

L. Waihman2002

3 4 or or x , etc.f x x x

radical

radicand

index

A radical function is a function that has a variable in the radicand.

3

4

y x

y x

y x

1 y x Horizontal shift left one

Vertical stretch two.

Vertical shift down three

You can apply the same transformations to the graphs of radical functions as you can to polynomial functions.

3 y x

2y x

Radical Parent functionsy x 3y x 4y x

0,0

1,1

4,2

9,3

8, 2

1, 1

0,0

1,1

8,2

0,0

1,1

16,2

81,3

As x, f(x)As x0+, f(x)0 As x, f(x)

As x–, f(x)–

As x, f(x)As x0+, f(x)0

Click on each graph above to link to interactive page!

Transformations

3y x 2y x 3y x

Vertical stretch of 3 Vertical shift up 2 Horizontal shift right 3

We apply the transformations on these functions in the samemanner as we did with the polynomial functions.

Domain:

Range

Domain:

Range:

Domain:

Range:

0,

0,

0,

2,

3,

0,

Applying the transformations

12( ) 2 4 f x x

Vertical shrink of ½

Vertical stretch 3

3 1 1 f x x

Horizontal shift right 2

Vertical shift up 4

Horizontal shift left 1

Vertical shift down 1Domain: 1, Range: 1, As x , f(x) ;

As x -1+, f(x) -1

As x , f(x) - ; As x 2+, f(x) 4Domain: Range: 2, ,4

Reflect @ x-axis

Applying the transformations 33 2 4 f x x

Vertical stretch of 3Horizontal shift left 2Vertical shift down 4

312 1 2 g x x

Vertical shrink of ½ Horizontal shift right 1Vertical shift up 2

Domain: Range: As x , f(x) ;As x - , f(x) -

, ,

Domain: Range:As x , f(x) - ;As x - , f(x)

Reflect @ x-axis

, ,

•To solve a radical equation that has only one variable in the radicand, isolate that term on one side of the equation. If the index is 2, then square both sides of the equation.

Be careful! The new equation you created when you Squared both sides might have extraneous solutions!

Isolate the radical.Square both sides.

Simplify and set equal to zero.Factor.Solve.

Radical Equations

6 14x x

6 14x x

26 196 28x x x 2 29 190 0x x

19 10 0x x

19 10 x or x

Given:

2 2

6 14x x

These solutions may not be solutions to the original equation.

?

6 14

19 6 19 14

24 14

x x

?

6 14

10 6 10 14

14 14

x x

Check your solutions!

The graph below illustrates thatthe derived equation may bedifferent from the originalequation.

√

10

Solving Radical Equations

Try: 10 12x x

10 12x x 210 24 144x x x 20 25 154x x

0 ( 11)( 14)x x

11 14x or x

Check:?

11 10 11 12

Try: 10 2x x

10 2x x 210 4 4x x x 20 5 6x x

0 6 1x x

6 1x or x

Check

?14 10 14 12

10 12

12 12

?1 10 ( 1) 2

3 ( 1) 2

?6 10 6 2

4 6 2

14

6

A radical equation may contain two radical expressions with an index of 2.To solve these, rewrite the equation with one of the radicals isolated on one side of the equals sign.Then, square both sides.If a variable remains in a radicand, you must repeat the squaring process.

More Solving Radical Equations

8 2 x x

8 2x x

8 4 4x x x

3 x

9 x

Check: 8 2 x x?

9 8 9 2 2 2

2 2

8 2x x

Try:

12 4 x

223 x

9

More Solving Radical Equations

Isolate one radical on each side of equals sign.

Square both sides.

Collect like terms on each side of equals sign.

Simplify.

Square both sides again.

More Solving Radical Equations

4 5 8x x 4 5 8x x 4 5 16 5 64x x x

4 60 16 5x x 15 4 5x x

2 30 225 80x x x 2 50 225 0x x

5 45 0x x 5 45x or x

Check:

?45 4 5(45) 8

7 15 8

?5 4 5(5) 8

3 5 8 √

5

Try:

Radical equations with indexes greaterthan 2 can be solved using similar techniques.

3 3 2 2 0x 3 3 2 2x

333 3 2 2x

3 2 8x 3 6x

2x

Check: ?

3 3 2 2 2 0 ?

3 8 2 0 2 2 0

After isolating the term containing the radical, raise each side of the equation to the power equal to the index of the radical.

√

2

Bonus Questions!One type of transformation was not covered in this PowerPoint. Identify it and give and example of an equation with this type of transformation and then sketch its graph to illustrate the transformation. Identify the domain and range as well. Solve the equation:

31 1 13x x x

click

3

333

2 3

2 3 2

3 2

3 2

1 1 13

1 1 13

1 3 3 1 13

1 3 3 14 13

2 11 12 0

2 11 12 0

3 1 4 0

3,1,4

x x x

x x x

x x x x x

x x x x x

x x x

x x x

x x x