UNIT 6.8 UNIT 6.8 GRAPHING GRAPHING

RADICALRADICALFUNCTIONSFUNCTIONS

Warm Up

Identify the domain and range of each function.

D: R; R:{y|y ≥2}1. f(x) = x2 + 2

D: R; R: R2. f(x) = 3x3

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

g(x) =(x + 2)24. f is translated 2 units left.

Graph radical functions and inequalities.

Transform radical functions by changing parameters.

Objectives

radical functionsquare-root function

Vocabulary

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.

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 .

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

x (x, f(x))–8 (–8, –2)

–1 (–1,–1)

0 (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.

Check It Out! Example 1a Continued

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}.

•

••

•

Check It Out! Example 1bGraph 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.

Translate f 1 unit up. ••

Check It Out! Example 2a

g(x) = x + 1

f(x)= x

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

Check It Out! Example 2b

g is f vertically compressed

by a factor of .1

2

f(x) = x

Transformations of square-root functions are summarized below.

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

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

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

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

Special airbags are used to protect scientific equipment when a rover lands on the surface of Mars. On Earth, the function f(x) = approximates an object’s downward velocity in feet per second as the object hits the ground after bouncing x ft in height.

Check It Out! Example 5

The downward velocity function for the Moon is a horizontal stretch of f by a factor of about . Write the velocity function h for the Moon, and use it to estimate the downward velocity of a landing craft at the end of a bounce 50 ft in height.

25 4

64x

Step 2 Find the value of g for a bounce of 50ft.

Substitute 50 for x and simplify.

The landing craft will hit the Moon’s surface with a downward velocity of about 23 ft at the end of the bounce.

Check It Out! Example 5 ContinuedStep 1 To compress f horizontally by a factor of ,

multiply f by .

25 4 4

25

4 25625 25

425

h(x) = f x = • 64x =

h (x) = • ≈25623

2550

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

Graph the inequality.

x –4 –3 0 5

y 0 1 2 3

Check It Out! Example 6a

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

y = x+4

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

Check It Out! Example 6a Continued

Graph the inequality.

x –4 –3 0 5

y 0 1 2 3

Check It Out! Example 6b

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

3y = x − 3

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

Check It Out! Example 6b Continued

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