Holt McDougal Algebra 2

Radical Functions

Graph radical functions and inequalities.

Transform radical functions by changing parameters.

Objectives

Holt McDougal Algebra 2

Radical Functions

radical functionsquare-root function

Vocabulary

Holt McDougal Algebra 2

Radical Functions Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs below show the inverses of the quadratic parent function and cubic parent function.

Holt McDougal Algebra 2

Radical Functions

Notice that the inverses of f(x) = x2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x2 to x ≥ 0, its inverse is the function .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 .

Holt McDougal Algebra 2

Radical Functions

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.

Holt McDougal Algebra 2

Radical Functions Example 1A Continued

x (x, f(x))347

12

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

Holt McDougal Algebra 2

Radical Functions

Graph each function and identify its domain and range.

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.

Holt McDougal Algebra 2

Radical Functions

x (x, f(x))–6 1

23

10

Example 1B Continued

The domain is . The range is .

Holt McDougal Algebra 2

Radical Functions Example 1B Continued

Check Graph the function on a graphing calculator.

Holt McDougal Algebra 2

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.

Check It Out! Example 1a

Holt McDougal Algebra 2

Radical Functions

x (x, f(x))–8 –1

018

The domain is . The range is .

Check It Out! Example 1a Continued

Holt McDougal Algebra 2

Radical Functions

Check Graph the function on a graphing calculator.

Check It Out! Example 1a Continued

Holt McDougal Algebra 2

Radical Functions

x (x, f(x))–138

15

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

Check It Out! Example 1bGraph each function, and identify its domain and range.

Holt McDougal Algebra 2

Radical Functions

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.

Holt McDougal Algebra 2

Radical Functions

Holt McDougal Algebra 2

Radical Functions

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

Example 2: Transforming Square-Root Functions

g(x) = x + 5

f(x) = x

Holt McDougal Algebra 2

Radical Functions

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

Check It Out! Example 2a

g(x) = x + 1

f(x)= x

Holt McDougal Algebra 2

Radical Functions

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

Check It Out! Example 2b

f(x) = x

Holt McDougal Algebra 2

Radical Functions

Transformations of square-root functions are summarized below.

Holt McDougal Algebra 2

Radical Functions Example 3: Applying Multiple Transformations

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

f(x)= x

Holt McDougal Algebra 2

Radical Functions 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

Holt McDougal Algebra 2

Radical Functions Check It Out! Example 3a

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

f(x)= x

Holt McDougal Algebra 2

Radical Functions

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

●

●

Check It Out! Example 3a

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

f(x)= x

Holt McDougal Algebra 2

Radical Functions 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

Holt McDougal Algebra 2

Radical Functions

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

Holt McDougal Algebra 2

Radical Functions Example 4: Writing Transformed Square-Root

FunctionsUse 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

Holt McDougal Algebra 2

Radical Functions

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

Holt McDougal Algebra 2

Radical Functions

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

Holt McDougal Algebra 2

Radical Functions Example 6: Graphing Radical Inequalities

x 0 1 4 9y

Graph the inequality .

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

y =2 x 3

Holt McDougal Algebra 2

Radical Functions

Graph the inequality.

x –4 –3 0 5y

Check It Out! Example 6a

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

y = x+4

Holt McDougal Algebra 2

Radical Functions

Graph the inequality.

x –4 –3 0 5y

Check It Out! Example 6b

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

3y x 3

Holt McDougal Algebra 2

Radical Functions Lesson Quiz: Part I

D:{x|x }; R:{y|y }

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

Holt McDougal Algebra 2

Radical Functions Lesson Quiz: Part II

2. Using the graph of as a guide, describe the transformation and graph the function . gx = x + 3

Holt McDougal Algebra 2

Radical Functions Lesson Quiz: Part III

3. Graph the inequality .