Radical Functions 8-7 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Vocabulary radical function square-root function

A radical function is a function whose rule is a radical expression 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.

x (x, f(x)) Example 1A Continued 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}.

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.

x (x, f(x)) Example 1B Continued –6 (–6, –4) 1 (1,–2) 2 (2, 0) 3 (3, 2) 10 (10, 4) ● ● ● ● ● The domain is the set of all real numbers. The range is also the set of all real numbers

Example 1B Continued Check Graph 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 (x, f(x)) –8 (–8, –2) –1 (–1,–1) (0, 0) 1 (1, 1) 8 (8, 2) • • • • • 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.

• • • • x (x, f(x)) Check It Out! Example 1b Graph each function, and identify its domain and range. 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}.

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.

Example 2: Transforming Square-Root Functions Using the graph of as a guide, describe the transformation and graph the function. f(x) = x g(x) = x + 5 • Translate f 5 units up.

Check It Out! Example 2a Using the graph of as a guide, describe the transformation and graph the function. f(x)= x • g(x) = x + 1 Translate f 1 unit up.

g is f vertically compressed by a factor of . Check It Out! Example 2b Using the graph of as a guide, describe the transformation and graph the function. f(x) = x g is f vertically compressed by a factor of . 1 2

Transformations of square-root functions are summarized below.

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.

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.

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

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 1 5 Step 1 Identify how each transformation affects the function. Reflection across the x-axis: a is negative 1 5 a = – Vertical compression by a factor of 1 5 Translation 5 units down: k = –5

Step 2 Write the transformed function. Example 4 Continued Step 2 Write the transformed function. 1 5 g(x) = - x + (- 5) ö ÷ ø æ ç è Substitute – for a and –5 for k. 1 5 Simplify.

Example 4 Continued Check Graph both functions on a graphing calculator. The g indicates the given transformations of f.

Check It Out! 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! Example 4 Continued Step 2 Write the transformed function. Substitute –2 for a and 1 for k. Simplify. Check Graph 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.

Example 6: Graphing Radical Inequalities Graph the inequality . Step 1 Use the related equation to make a table of values. y =2 x -3 x 1 4 9 y –3 –1 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 

x –4 –3 5 y 1 2 3 Check It Out! Example 6a Graph the inequality. Step 1 Use the related equation to make a table of values. y = x+4 x –4 –3 5 y 1 2 3

Check It Out! 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! 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 

x –4 –3 5 y 1 2 3 Check It Out! Example 6b Graph the inequality. Step 1 Use the related equation to make a table of values. 3 y = x - 3 x –4 –3 5 y 1 2 3

Check It Out! 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! Example 6b Continued Check Choose a point in the solution region, such as (4, 2), and test it in the inequality. 2 ≥ 1 

Lesson Quiz: Part I 1. Graph the function and identify its range and domain. D:{x|x≥ –4}; R:{y|y≥ 0} •

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 3. Graph the inequality .