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SQUARE ROOT SQUARE ROOT Functions Functions

Radical functionsRadical functions

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REVIEWREVIEWA radical function is a function whose rule is a radical expression, which include the square-root function, x

f x x

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EXAMPLE 1EXAMPLE 1Graph each function and identify its domain and range of 2 1f x x

1x 0y Domain: Range:

g(x)x f(x) 2 1f x x

0246

2 1 1 2 0 12 0 1 2 1

2 3 1 2 2

2 8 1 2 3

0

38

1x x 0y y

0, 1,

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EXAMPLE 2EXAMPLE 2Graph each function and identify its domain and range of 2 1f x x

Domain: Range:0x 1y 0x x 0,

1y y

,1

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EXAMPLE 3EXAMPLE 3Using the graph, f(x) = , as a guide, describe the transformation, identify the domain and range, and graph the function, 4g x x

Domain:

Range:

4x

0y g(x)

g(x) translates 4 units right

x

4x x

0y y 4,

0,

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EXAMPLE 4EXAMPLE 4Using the parent function as a guide, describe the transformation, identify the domain and range, and graph the function, 4g x x

Domain: Range:0x 4y

g(x)

g(x) translates 4 units down

0x x

0, 4y y

4,

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YOUR TURNYOUR TURNUsing the parent function as a guide, describe the transformation, identify the domain and range, and graph the function, 5 5g x x

Domain: Range:

5x 5y

g(x)

g(x) translates 5 units left and 5 units down

5x x

5, 5,

5y y

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YOUR TURNYOUR TURNGraph each function and identify its domain and range of 2 1 3f x x

Domain:

Range:

1,

3,

g(x) translates 1 unit right, 3 units up and stretches by a factor of 2

1x

3y

1x x

3y y

Example 3: Applying Multiple Transformations

Reflect f across the x-axis, and translate it 4 units to the right.

••

Using the graph of as a guide, describe the transformation and graph the function .

f(x)= x

Lesson Quiz: Part II

g is f reflected across the y-axis and translated 3 units up.

Using the graph of as a guide, describe the transformation and graph the function .

•

•

gx = x + 3

g is f vertically stretched by a factor of 3, reflected across the x-axis, and translated 1 unit down.

●●

Check It Out! Example 3b

Using the graph of as a guide, describe the transformation and graph the function.

f(x)= x

g(x) = –3 x – 1

Example 4: Writing Transformed Square-Root Functions

Use the description to write the square-root

function g. The parent function is

reflected across the x-axis, compressed vertically

by a factor of , and translated down 5 units.

15

f(x)= x

Use the description to write the square-root function g.

Check It Out! Example 4

The parent function is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.

f(x)= x

Example 6: Graphing Radical Inequalities

x 0 1 4 9

y

Graph the inequality .

Step 1 Use the related equation to make a table of values.

y =2 x 3

Example 6 Continued

Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.

Because the value of x cannot be negative, do not shade left of the y-axis.

Example 6 Continued

Check Choose a point in the solution region, such as (1, 0), and test it in the inequality.

0 > 2(1) – 3

0 > –1

Graph the inequality.

x –4 –3 0 5

y 0 1 2 3

Check It Out! Example 6a

Step 1 Use the related equation to make a table of values.

y = x+4

Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.

Because the value of x cannot be less than –4, do not shade left of –4.

Check It Out! Example 6a Continued

Check Choose a point in the solution region, such as (0, 4), and test it in the inequality.

4 > (0) + 4

4 > 2

Check It Out! Example 6a Continued

Graph the inequality.

x –4 –3 0 5

y 0 1 2 3

Check It Out! Example 6b

Step 1 Use the related equation to make a table of values.

3y x 3

Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.

Check It Out! Example 6b Continued

Check Choose a point in the solution region, such as (4, 2), and test it in the inequality.

2 ≥ 1

Check It Out! Example 6b Continued

Lesson Quiz: Part I

D:{x|x≥ –4}; R:{y|y≥ 0}

1. Graph the function and identify its range and domain.

•

Lesson Quiz: Part II

g is f reflected across the y-axis and translated 3 units up.

2. Using the graph of as a guide, describe the transformation and graph the function .

•

•

gx = x + 3

Lesson Quiz: Part III

3. Graph the inequality .

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