holt algebra 2 8-7 radical functions 8-7 radical functions holt algebra 2 warm up warm up lesson...

Holt Algebra 2

8-7 Radical Functions 8-7 Radical Functions

Holt Algebra 2

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Lesson QuizLesson Quiz

Holt Algebra 2

8-7 Radical Functions

radical functionsquare-root function

Vocabulary

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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 .

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Graph each function and identify its domain and range.

Example 1A: Graphing Radical Functions

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.

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Example 1A Continued

x (x, f(x))3 (3, 0)

4 (4, 1)

7 (7, 2)

12 (12, 3)

The domain is {x|x ≥3}, and the range is {y|y ≥0}.

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Example 1B: Graphing Radical Functions

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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x (x, f(x))–6 (–6, –4)

1 (1,–2)

2 (2, 0)

3 (3, 2)

10 (10, 4)

Example 1B Continued

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

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Example 1B Continued

Check Graph the function on a graphing calculator.

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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

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x (x, f(x))–8 (–8, –2)

–1 (–1,–1)

0 (0, 0)

1 (1, 1)

8 (8, 2)

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• •

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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

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Check Graph the function on a graphing calculator.


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x (x, f(x))–1 (–1, 0)

3 (3, 2)

8 (8, 3)

15 (15, 4)

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

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Check It Out! Example 1b

Graph each function, and identify its domain and range.

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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.

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Holt Algebra 2


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

Example 2: Transforming Square-Root Functions

Translate f 5 units up.

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g(x) = x + 5

f(x) = x

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Translate f 1 unit up. ••


g(x) = x + 1

f(x)= x

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g is f vertically compressed

by a factor of .1

2

f(x) = x

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Transformations of square-root functions are summarized below.

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Example 3: Applying Multiple Transformations

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

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

f(x)= x

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

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f(x)= x

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

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f(x)= x

g(x) = –3 x – 1

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Example 4: Writing Transformed Square-Root Functions

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

Translation 5 units down: k = –5

Vertical compression by a factor of 15

15

a = –

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

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Example 4 Continued

Step 2 Write the transformed function.

Simplify.

Substitute – for a and –5 for k.15

15

g(x) = x + ( 5)

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Example 4 Continued

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

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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.

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

Translation 5 units down: k = 1

Vertical compression by a factor of 2 a = –2

f(x)= x

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Simplify.

Substitute –2 for a and 1 for k.

Step 2 Write the transformed function.

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

Check It Out! Example 4 Continued

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

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Example 6: Graphing Radical Inequalities

x 0 1 4 9

y –3 –1 1 3

Graph the inequality .

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

y =2 x 3

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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.

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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

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Graph the inequality.

x –4 –3 0 5

y 0 1 2 3



y = x+4

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


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4 > (0) + 4

4 > 2


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Graph the inequality.

x –4 –3 0 5

y 0 1 2 3



3y x 3

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Check It Out! Example 6b Continued

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2 ≥ 1

Check It Out! Example 6b Continued

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Lesson Quiz: Part I

D:{x|x≥ –4}; R:{y|y≥ 0}

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

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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 .

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gx = x + 3

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Lesson Quiz: Part III

3. Graph the inequality .

holt algebra 2 8-7 radical functions 8-7 radical functions holt algebra 2 warm up warm up lesson...

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