Warm Up Identify the domain and range of each function. 1. f(x) = x2 + 2 D: R; R:{y|y ≥2} 2. f(x) = 3x3 D: R; R: R Use the description to write the quadratic function g based on the parent function f(x) = x2. 3. f is translated 3 units up. g(x) = x2 + 3 4. f is translated 2 units left. g(x) =(x + 2)2

Objectives Graph radical functions and inequalities. Transform radical functions by changing parameters.

Vocabulary radical function square-root function

Quadratic and cubic functions have inverses Quadratic and cubic functions have inverses. The graphs below show the inverses of the quadratic parent function and cubic parent function.

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 .

Example 1A: Graphing Radical Functions Graph the function and identify its domain and range.

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 the function and identify its domain and range.

Check It Out! Example 1a Graph the function and identify its domain and range.

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

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.

Using the graph of as a guide, describe the transformation and graph the function. f(x) = x g(x) = x + 5

Using the graph of as a guide, describe the transformation and graph the function. f(x)= x g(x) = x - 1

Using the graph of as a guide, describe the transformation and graph the function. f(x) = x

Using the graph of as a guide, describe the transformation and graph the function. f(x) = x  

Using the graph of as a guide, describe the transformation and graph the function. f(x) = x  

Using the graph of as a guide, describe the transformation and graph the function. f(x) = x  

Example 3: Applying Multiple Transformations Using the graph of as a guide, describe the transformation and graph the function f(x)= x .