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A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving . The square-root parent function is . The cube-root parent function is .

Example 1A: Graphing Radical Functions

Graph each function and identify its domain and range.

Make a table of values. Plot enough ordered pairs to see the shape of the curve. Because the square root of a negative number is imaginary, choose only nonnegative values for x – 3.

Example 1B: Graphing Radical Functions

Graph each function and identify its domain and range.

Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

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The domain is the set of all real numbers. The range is also the set of all real numbers

CheckGraph the function on a graphing calculator.

Check It Out! Example 1a

Graph each function and identify its domain and range.

Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

Check It Out! Example 1a Continued

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The domain is the set of all real numbers. The range is also the set of all real numbers.

Check It Out! Example 1a Continued

Check Graph the function on a graphing calculator.

Check It Out! Example 1b

Graph each function, and identify its domain and range.

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The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

g(x) = x + 5 methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

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Example 2: Transforming Square-Root Functions

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

f(x) = x

Translate f5 units up.

g(x) = x + 1 methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

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Check It Out! Example 2a

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

f(x)= x

Translate f1 unit up.

g methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions. is f vertically compressed by a factor of .

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Check It Out! Example 2b

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

f(x) = x

• below.

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Example 3: Applying Multiple Transformations

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

f(x)= x

.

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

● below.

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Check It Out! Example 3a

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

f(x)= x

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

g(x) = –3 x – 1 below.

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Check It Out! Example 3b

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

f(x)= x

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

a below. = –

1

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1

Vertical compression by a factor of

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5

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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.

f(x)= x

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

Translation 5 units down: k = –5

1 below.

ö

æ

g(x) =- x + (-5)

Substitute – for a and –5 for k.

÷

ç

5

ø

è

1

5

Example 4 Continued

Step 2 Write the transformed function.

Simplify.

Example 4 Continued below.

CheckGraph both functions on a graphing calculator. The g indicates the given transformations of f.

Check It Out! below. Example 4

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

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

f(x)= x

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

a = –2

Vertical compression by a factor of 2

Translation 5 units down: k = 1

Check It Out! below. Example 4 Continued

Step 2 Write the transformed function.

Substitute –2 for a and 1 for k.

Simplify.

CheckGraph both functions on a graphing calculator. The g indicates the given transformations of f.

In addition to graphing radical functions, you can also graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

Step 1 graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Use the related equation to make a table of values.

y =2 x -3

Example 6: Graphing Radical Inequalities

Graph the inequality .

Example 6 Continued graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

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 graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

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

0 > 2(1) – 3

0 > –1

Step 1 graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Use the related equation to make a table of values.

y = x+4

Check It Out! Example 6a

Graph the inequality.

Check It Out! graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Example 6a 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 less than –4, do not shade left of –4.

Check It Out! graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. 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

Step 1 graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Use the related equation to make a table of values.

y = x - 3

3

Check It Out! Example 6b

Graph the inequality.

Check It Out! graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Example 6b 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.

Check It Out! graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities. Example 6b Continued

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

2 ≥ 1

1 graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.. Graph the function and identify its range and domain.

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Lesson Quiz: Part I

D:{x|x≥ –4}; R:{y|y≥ 0}

• graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

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Lesson Quiz: Part II

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

g(x) = -x + 3

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

Lesson Quiz: Part III graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

3. Graph the inequality .